Ring Theory Abstracts

Almost injectivity

ADEL ALAHMADI, King Abdulaziz University

This is a brief survey on a concept called almost injectivity that has been studied mostly by Harada, and his collaborators. Baba introduced this concept as a dual to the concept of almost projectivity. In this survey, we state salient results by main contributors on this topic and some recent results by the authors. This exposition is not claimed to be exhaustive but is meant to highlight important discoveries made in the area and to point out future possible work. A list of open questions is given. The interested readers, particularly graduate students would benefit most by going over the wealth of information contained in the referenced articles on almost injectivity and projectivity (Joint with S. K. Jain, Ohio University and King Abdulaziz University)

KING ABDULAZIZ UNIVERSITY
JEDDAH, SAUDI ARABIA
adelnife2 at yahoo dot com

On a generalization of extending modules by torsion theory

MUSTAFA ALKAN, Akdeniz University

We introduce a generalization of extending (CS) modules by using the concept of $\tau$ -large submodule which was defined in [1]. We give some properties of this class of modules and study their relationship with the familiar concepts of $\tau$ -closed, $\tau$ -complement submodules and the other generalization of extending modules ($\tau$ -complemented, $\tau$ -CS, s-$\tau$ -CS modules). We are also interested in determining when a $\tau$ -divisible module is $\tau$ -extending. For a $\tau$ -extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that M=??? and K is $\tau$ (M)-injective. We also treat when a direct sum of -extending modules is $\tau$ -extending. Keywords: $\tau$ -large submodule, $\tau$ -closed submodule, extending module, $\tau$ -complemented module, $\tau$ -CS module.

References [1]-Gomez Pardo, J. L. (1985). Spectral Gabriel Topologies and Relative Singular Functors. Communications in Algebra. 13 (1): 21-57.

ANTALYA, TURKEY
alkan at akdeniz dot edu dot tr

The enumeration of finite and artinian chain rings

YOUSEF ALKHAMEES, King Saud University

We consider only associative right and left artinian rings with identity. A chain ring is a ring whose left (right) ideals form a chain. Let R be a ring, then R is a local principal ring if and only if it is a chain ring. The motivation for the increased interest in chain rings is the recent use of finite chain rings in coding theory. We associate with each finite (artinian) chain ring six invariants(integers) and determine the number of isomorphism classes of chain rings with given invariants in terms of the number of commutative chain rings. Also we determine the enumeration of finite and artinian chain rings under some suitable conditions.

KING SAUD UNIVERSITY
RIYADH, SAUDI ARABIA
ykhamees at gmail dot com

Leavitt path algebras of finite Gelfand-Kirillov dimension

HAMED ALSULAMI, King Abdulaziz University

Let $\Gamma$ be a finite graph and let $L(\Gamma )$ denote the Leavitt path algebra. Let $C^{\prime }$ and $C^{\prime \prime }$ be two cycles. We write $C^{\prime }\Rightarrow C^{\prime \prime }$ if there is a path that starts from $C^{\prime }$ and finishes at $C^{\prime \prime }$ . Clearly, under the hypothesis that no two cycles have a common vertex, if $% C^{\prime }\Rightarrow C^{\prime \prime }$ and $C^{^{\prime \prime }}\Rightarrow C^{\prime }$ then $C^{\prime }=C^{\prime \prime }.$ If $C_{1}$ , $C_{2}$ ,.....,$C_{n}$ are distinct cycles and $C_{1}\Rightarrow C_{2}\Rightarrow C_{3}$ ..... $\Rightarrow C_{n}$ then we call this a chain of length $n.$ We say a chain has an exit if $C_{n}$ has an exit. Let $d_{1}$ be a maximal length of a chain of cycles with an exit and let $d_{2}$ be a maximal length of a chain of cycles without an exit.

Theorem 1. $GK$ $\dim$ $L(\Gamma )$ is finite if and only if no two different cycles have a common vertex.

Theorem 2. $GK$ $\dim$ $L(\Gamma )$ = $max(2d_{1},2d_{2}-1)$ or $\infty$

(joint work with Adel Alahmadi, S. K. Jain and Efim Zelmanov)

KING ABDULAZIZ UNIVERSITY
JEDDAH, MEKKAH, SAUDI ARABIA
hamed9 at hotmail dot com

Group developed weighing matrices

K.T. ARASU, Wright State University

Let $H$ be a group of order $n$ ($H$ need not be abelian, but we write $H$ additively). An $n \times n$ matrix $A=(a_{gh} )$ indexed by the elements of the group $H$ (so $g$ and $h$ belong to $H$ ) is said to be $H$ -developed (or $H$ -invariant) if it satisfies the condition $a_{gh}=a_{g+k,h+k}$ for all $g,h,k\in H$ .

$A$ is said to be circulant if the underlying group $H$ is cyclic. Thus the matrix $A$ is completely determined by its first row. Let RG denote the group ring of a given group G over a ring R. Then the set of G-invariant matrices with entries from R is isomorphic to the group ring RG. We report on some recent results on the following classes of objects that arise from group invariant matrices: Group weighing matrices and perfect arrays with complex roots of unity as its entries and sequences with ideal autocorrelations.

WRIGHT STATE UNIVERSITY
DAYTON, OH
k.arasu at wright dot edu

Recent results in skew cyclic codes

NUH AYDIN, Kenyon College

Skew cyclic codes, also called $\theta$ -cyclic codes, are a recently introduced generalization of ordinary cyclic codes. Unlike most other codes related to cyclic codes, one needs to work in the unfamiliar territory (for a coding theorist!) of the non-commutative ring of skew polynomial ring over a field which does not possess the unique factorization property. There have been useful results and new codes obtained from this class of codes and their generalizations. This talk will give an overview of recent results in the field.

KENYON COLLEGE
GAMBIER, OH
aydinn at kenyon dot edu

Rings whose simple modules have maximal or

minimal injectivity domains

PINAR AYDOGDU, Hacettepe University

In a recent paper of Alahmadi, Alkan and López-Permouth, a ring $R$ is defined to have no simple middle class if the injectivity domain of any simple $R$ -module is the smallest or largest possible. In this talk, we discuss some properties of rings with no simple middle class. It is shown that if $R$ is a right Artinian ring, then $R$ has no simple middle class if and only if there is a ring decomposition $R=S\oplus T$ , where $S$ is semisimple Artinian and $T$ is zero or has one of the following properties:

$(1)$ $T$ is a right $SI$ -ring with homogeneous right socle.

$(2)$ $T$ has a unique noninjective simple right $R$ -module up to isomorphism, and the right socle of $T$ is (homogeneous) singular.

Furthermore, if $R$ is a commutative Noetherian ring, then $R$ has no simple middle class if and only if there is a ring decomposition $R=oplus T$ , where $S$ is semisimple Artinian and $T$ is a local ring.
(This is a joint work with Bülent Saraç).

HACETTEPE UNIVERSITY
ANKARA, TURKEY
paydogdu at hacettepe dot edu dot tr

Partially balanced incomplete block designs with

two associate classes via additive groups of a finite field

PRADEEP BANSAL, Indian Institute Of Technology Guwahati

We provide constructions of cyclic 2-class Partially Balanced Incomplete Block Designs using cyclotomy in finite fields. Our results give theoretical explanation of the two sporadic examples given by Agrawal (1987). This is a joint work with K.T.Arasu and Cody Watson.

INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI
GUWAHATI, ASSAM, INDIA
bpradeep20 at gmail dot com

On commutative weakly FC-rings

MAMADOU BARRY, Cheikh Anta Diop University Dakar ( Senegal )

Let R be a commutative ring and M an unital R-module. M is called co-Hopfian if any injective endomorphism of M is an isomorphism. M is called weakly co-Hopfian if any injective endomorphism of M is essential. The ring R is called weakly FC-ring if any weakly co-Hopfian R-module if finitely cogenerated. In this paper, we show that for a commutative ring R, the following conditions are equivalent : 1 - R is a weakly FC-ring. 2 - R is an Artinian principal ideal ring. Definitions and notations used in this paper can be found in [1] and [10]. Keywords: weakly FC-ring, Artinian principal ideal ring, finitely generated module, finitely cogenerated module, co-Hopfian module, weakly co-Hopfian module, uniform dimension.

References

[1] F. W. Anderson and K. R. Fuller : Rings and categories of modules, New York Springer-Verlag, Berlin, 1973. [2] SH. Asgary On weakly co-Hopfian modules. Bulletin of Iranian Mathematical Society. Vol. 33 N 1 (2007) pp. 65 - 72. [3] Barry, M. Diankha, O., Sangharé, M. : On commutative FGI-rings. Math. Sc. Res. J.9 (4) (2005) 87 - 91. [4] Barry, M., P.C. Diop :Some properties related to commutative weakly FGI-rings, J P Journal of Algebra, Number Theory and application Vol.19, issue 2,(2010) 141-153. [5] I. S. Cohen : On the structure and ideal theory of complete local rings, Trans. Amer. Soc., 59 (1946), 54 - 106. [6] I. S. Cohen, I. Kaplansky : Ring for which every module is a direct sum of cyclic modules, Math. Zeitschr Bd., 54(H2S), 97 - 101. [7] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer : Extending Modules, Longman : Burnt Mill, 1994. [8] A. Haghany, M. R. Vedadi : Modules whose injective endomorphism are essential. J. Algebra 243 (2001) 765 - 779. [9] M. A. Kaïdi, M. Sangharé : Une caractérisation des anneaux artiniens à idéaux principaux in L.N.M., vol. 1328, Springer-Verlag, Berlin, 1988, 245 - 254. [10] T. Y. Lam : Lectures on modules and rings. G. T. M. (189), Springer- Verlag. Berlin-Heidelberg, New York(1999). [11] Sharpe, D.W. and Vamos, P. : Injective modules. Cambridge University Presss (1972). [12] W. V. Vasconscelos : On finitely generated flat modules. Proc. Am. Math. Soc. 138 (1969) 900 - 901. [13] W. V. Vasconscelos : Injective endomorphism of finitely generated modules. Proc. Am. Math. Soc. 25(1970) 505 - 512.

CHEIKH ANTA DIOP UNIVERSITY DAKAR ( SENEGAL )
DAKAR, SENEGAL
mansabadion1 at hotmail dot com

Thoughts on Eggert's Conjecture

GEORGE MARK BERGMAN, University of California, Berkeley

Eggert's Conjecture says that if one applies to a finite-dimensional nilpotent commutative algebra $R$ over a perfect field of characteristic $p$ the $p$ -th-power map, then the dimension of the algebra will shrink by a factor of at least $p$ . Whether this elementary statement is true is not known! I will motivate the conjecture, and discuss versions of it that are not limited to prime characteristic and/or to commutative $R$ , consequences it would have for finite semigroups, and some examples.

UNIVERSITY OF CALIFORNIA, BERKELEY
BERKELEY, CALIFORNIA
gbergman at math dot berkeley dot edu

Invariance and the extending condition

GARY F. BIRKENMEIER, University of Louisiana at Lafayette

Let C be a subset of the set of all submodules of a module M. We say M is "C-extending" if every element of C is essential in a direct summand of M. When H is a nonempty subset of $End(M_R)$ and C is the set of submodules of M which are invariant with respect to H, we say M is HI-extending. The C-extending and HI-extending conditions are investigated with respect to forming direct sums, direct summands, and dense extensions. Applications are made when H is the set of all idempotents of $End(M_R)$ , then C is the set of projective invariant submodules of M. The projective invariant extending Abelian groups are characterized. Examples are presented to illustrate and delimit our results. (This is joint work with Adnan Tercan and Canan C. Yucel).

UNIVERSITY OF LOUISIANA AT LAFAYETTE
LAFAYETTE, LA
gfb1127 at louisiana dot edu

On elements in algebras having finite number of conjugates

VICTOR BOVDI, University of Debrecen

Let $R$ be a ring with unity and $U(R)$ its group of units. Let

$$\Delta U=a\in U(R)\mid [U(R):C_{U(R)}(a)]<\infty\}$$

be the $FC$ -radical of $U(R)$ and let

$$\nabla(R)=a\in R\mid [U(R):C_{U(R)}(a)]<\infty\}$$

be the $FC$ -subring of $R$ .

The investigation of the $FC$ -radical $\Delta U$ and the $FC$ -subring $\bigtriangledown(R)$ was proposed by H. Zassenhaus (see S. K. Sehgal and H. Zassenhaus, On the supercentre of a group and its ring theoretic generalization, In Integral representations and applications (Oberwolfach, 1980), volume 882 of Lecture Notes in Math., pages 117-144. Springer, Berlin, 1981; S. K. Sehgal and H. J. Zassenhaus, Group rings whose units form an FC-group, Math. Z., 153(1):29-35, 1977.) They described the $FC$ -subring of $\mathbb{Z}$ -order as a unital ring with a finite $\mathbb{Z}$ -basis and a semisimple quotient ring.

An infinite subgroup $H$ of $U(R)$ is said to be an $\omega$ -subgroup if the left annihilator of each nonzero Lie commmutator $[x,y]$ in $R$ contains only finite number of elements of the form $1-h$ , where $x,y \in R$ and $h\in H$ . In the case when $R$ is an algebra over a field $F$ , and $U(R)$ contains an $\omega$ -subgroup, we describe (see V. Bovdi, On elements in algebras having finite number of conjugates, Publ. Math. Debrecen, 57(1-2):231-239, 2000) its $FC$ -subalgebra and the $FC$ -radical. This result is an extension of V. Bovdi, Twisted group rings whose units form an ${\rm FC}$ -group, Canad. J. Math., 47(2):274-289, 1995; G. H. Cliff and S. K. Sehgal, Group rings whose units form an $FC$ -group, Math. Z., 161(2):163-168, 1978; S. K. Sehgal and H. Zassenhaus, On the supercentre of a group and its ring theoretic generalization, In Integral representations and applications (Oberwolfach, 1980), volume 882 of Lecture Notes in Math., pages 117-144. Springer, Berlin, 1981; S. K. Sehgal and H. J. Zassenhaus, Group rings whose units form an FC-group, Math. Z., 153(1):29-35, 1977.)

UNIVERSITY OF DEBRECEN
DEBRECEN,
vbovdi at gmail dot com

Counting cleanliness

VICTOR CAMILLO, U of Iowa

The purpose of this short talk is to raise a question. S is the nxn matrix ring over a field C Background: 1. Fitting;s Lemma asserts that every element in S is a sum of a unit and an idempotent that commute. 2. Let X be the 0,1 diagonal matrices, a straigtforward determinant induction argument shows that given a matrix M, one may subtract an element in X from M to obtain a unit. X is thus an "adequate" set of idempotents. Question, is it the case that every adequate set of idemotents must have n choose k idempotents of rank k. In particular must an adequate set have $2^n$ elements. The REU's have proved this up to n=

U OF IOWA
IOWA CITY, IA
camillo at math dot uiowa dot edu

On additive commutators in associative rings

MIKHAIL CHEBOTAR, Kent State University

We will briefly discuss some questions on additive commutators in associative rings. In particular, a solution of a problem by Herstein on commutators and nilpotent elements in simple rings (joint work with E.R. Puczylowski and P.-H. Lee) will be presented.

KENT STATE UNIVERSITY
KENT, OH
chebotar at math dot kent dot edu

A class of strongly clean rings

JIANLONG CHEN, Department of Mathematics, Southeast University

Motivated by results about an element of a ring that admits a spectral idempotent (so-called a quasipolar element in literature), we introduce notions of pseudopolar rings and quasipolar rings. A ring $R$ is called quasipolar if every element of $R$ is quasipolar (or, every element of $R$ has a generalized Drazin inverse). It is proved that strongly $\pi$ -regular rings and uniquely strongly clean rings are pseudopolar, pseudopolar rings are quasipolar, and quasipolar rings are strongly clean. In this talk, we present Cline's formula and Jacobson's lemma for the generalized Drazin inverse. The quasipolarity and pseudopolarity of matrix rings and the triangular matrix rings over local rings are also considered.

DEPARTMENT OF MATHEMATICS, SOUTHEAST UNIVERSITY
NANJING, JIANGSU PROVINCE
jlchen at seu dot edu dot cn

Galois coverings of coalgebras

WILLIAM CHIN, DePaul University

We introduce the concept of a Galois covering of a pointed coalgebra. By an analog of a fundamental result of Gabriel, a pointed coalgebra embeds into the path coalgebra of its quiver. Topological coverings of quivers are used to construct covering coalgebras, including a universal covering for a path subcoalgebra of the path coalgebra. The theory developed shows that Galois coverings of coalgebras can be expressed by smash coproducts using the coaction of the automorphism group of the covering. Thus the theory of Galois coverings of pointed coalgebras is seen to be equivalent to group gradings of coalgebras, and representations of coverings are equivalent to graded comodules. Gradings in connection to coverings of quivers and representation theory were studied in the 80's by Green, Martinez-de la Pena, Bongartz and Gabriel, Riedtman, and recently for k-categories by Cibils and Marcos. One feature of the coalgebra theory is that neither the grading group nor the quiver is assumed finite in order to obtain a smash product coalgebra.

DEPAUL UNIVERSITY
CHICAGO, IL
wchin at condor dot depaul dot edu

Stability of zero-divisor properties under

formation of the classical rings of Quotients

ALEXANDER J. DIESL, Wellesley College

In the commutative setting, many nice properties of a ring pass to its classical ring of quotients. In the noncommutative case, however, the situation is not nearly as nice. In this talk, we will focus on properties concerning zero-divisors, and we will present several constructions to illustrate the degree to which such properties pass from an Ore ring to its classical ring of quotients. This is joint work with C.Y. Hong, N.K. Kim and P.P. Nielsen.

WELLESLEY COLLEGE
WELLESLEY, MA
adiesl at wellesley dot edu

Some corollaries of Dimitric's

general characterization of slender objects

RADOSLAV DIMITRIC, CUNY

R. Dimitric gave an exhaustive characterization of slender objects in (Abelian) categories in early 1980's and subsequently. This work is given in considerable generality and as such may have not been understood well, so much so, that "rediscoveries" of partial results of this characterization seem to appear at steady rate. In this talk, I will give a few corollaries to my work, and especially in ring and module theory.

CUNY
NEW YORK, NY
RDIMITRIC at JUNO dot COM

Some results on torsionfree modules

NANQING DING, Department of mathematics, Nanjing University

We study torsionfree and divisible dimensions in terms of right derived functors of $-\otimes-$ . We also investigate the cotorsion pair cogenerated by the class of cyclic torsionfree right $R$ -modules. As applications, some new characterizations of von Neumann regular rings, $F$ -rings and semisimple Artinian rings are given. This talk is a report on joint work with Jiangsheng Hu.

DEPARTMENT OF MATHEMATICS, NANJING UNIVERSITY
NANJING, JIANGSU PROVINCE, CHINA
nqding at nju dot edu dot cn

On the structure of repeated-root constacyclic codes

of prime power length over a class of finite chain rings

HAI Q. DINH, Department of Mathematical Sciences, Kent State University.

The class of constacyclic codes plays a very significant role in the theory of error-correcting codes. In this talk, we consider repeated-root codes of prime power length over a class of finite chain rings, namely, ${\cal R}=\mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ , where $p$ is any prime. All constacyclic codes of length $p^s$ over the ring ${\cal R}$ are studied. The units of the chain ring ${\cal R}$ are of the forms $\gamma$ , and $\alpha+u\beta$ , where $\alpha, \beta, \gamma$ are nonzero elements of the Galois field $\mathbb{F}_{p^m}$ , which provides $p^m(p^m-1)$ such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length $p^s$ over the finite field $\mathbb{F}_{p^m}$ are obtained, and used as a tool to establish the structure and Hamming distances of all $(\alpha+u\beta)$ -constacyclic codes of length $p^s$ over ${\cal R}$ . We then classify all cyclic codes of length $p^s$ over ${\cal R}$ , and obtain the number of codewords in each of those cyclic codes. Finally, an one-to-one correspondence between cyclic and $\gamma$ -constacyclic codes of length $p^s$ over ${\cal R}$ is constructed via a ring isomorphism, that carries over the results about cyclic codes correspondingly to $\gamma$ -constacyclic codes of length $p^s$ over ${\cal R}$ .

DEPARTMENT OF MATHEMATICAL SCIENCES, KENT STATE UNIVERSITY.
WARREN, OHIO
hdinh at kent dot edu

Indecomposable modules over pure semisimple hereditary rings

NGUYEN VIET DUNG, Ohio University, Zanesville Campus

A ring $R$ is called left pure semisimple if every left $R$ -module is a direct sum of finitely generated modules, or equivalently, every left $R$ -module is pure-injective. It is still an open problem whether left pure semisimple rings always have finite representation type, known as the Pure Semisimplicity Conjecture. It is known that this conjecture can be reduced to the hereditary case. In this talk, we discuss some recent results on left pure semisimple hereditary rings. We describe the distribution of indecomposable left $R$ -modules over such a ring $R$ . We show, in particular, that every left pure semisimple hereditary ring has tame representation type, in the sense that for every natural number $n$ , there are only finitely many non-isomorphic indecomposable left $R$ -modules that belong to the Ziegler closure of the (finite) family of indecomposable left $R$ -modules of length $n$ . (This is joint work with José Luis García, University of Murcia, Spain)

OHIO UNIVERSITY, ZANESVILLE CAMPUS
ZANESVILLE, OHIO
nguyend2 at ohio dot edu

Descent of restricted flat Mittag-Leffler modules

SERGIO ESTRADA, Universidad de Murcia

Descent of a property of modules is a fundamental question in algebraic geometry when defining new classes of quasi-coherent sheaves. This means that the property can be defined using an open affine covering of the scheme. In the talk we will give an overview of the problem of descent with respect to several classes of modules that have been recently considered in [D,EGPT] to define a notion of (infinite dimensional) vector bundles.

The talk is part of a joint work with Pedro A. Guil Asensio and Jan Trlifaj.

References:

[D] V. DRINFELD, Infinite-dimensional vector bundles in algebraic geometry: an introduction, in The Unity of Mathematics, Birkhäuser, Boston 2006, pp. 263-304.

[EGPT] S. ESTRADA, P. GUIL ASENSIO, M. PREST, J. TRLIFAJ, Model category structures arising from Drinfeld vector bundles, preprint, arXiv:0906.5213.

UNIVERSIDAD DE MURCIA
MURCIA, SPAIN
sestrada at um dot es

On some noteworthy pairs of ideals in Mod-R

ALBERTO FACCHINI, University of Padova, Italy

We say that an additive functor $F\colon A \to B$ between preadditive categories $A$ and $B$ is a local functor if, for every morphism $f\colon X\to Y$ in $A$ , $F(f)$ isomorphism in $B$ implies $f$ isomorphism in $A$ . In this talk, we will show that there exists a number of pairs $(I_1, I_2)$ of ideals of $A$ for which the canonical functor $A \to A/I_1 \times A/I_1$ is a local functor. These pairs of ideals arise in a very natural way. Our main case will be when $A$ is the category Mod-$R$ of all right modules over a ring $R$ , or some other category strictly related to Mod-$R$ , like the full subcategory mod-$R$ of all finitely presented $R$ -modules, or the full subcategory of Mod-$R$ whose objects are all $R$ -modules with a projective cover.

UNIVERSITY OF PADOVA, ITALY
PADOVA, ITALY
facchini at math dot unipd dot it

Essentially regular algebras

JASON DALE GADDIS, University of Wisconsin - Milwaukee

An algebra is essentially regular of dimension $d$ if its homogenization is an Artin-Schelter regular algebra of dimension $d+1$ . We present a modified version of matrix congruence which can be used to classify algebras defined by generators and relations. This method can can be used to classify all 2-dimensional essentially regular algebras up to $\mathbb{K}$ -algebra isomorphism. We give this classification, along with that of the corresponding homogenized algebras.

UNIVERSITY OF WISCONSIN - MILWAUKEE
MILWAUKEE, WI
jdgaddis at uwm dot edu

Quantum matrices

KEN GOODEARL, University of California

The theme of the talk is aspects of quantum groups from a ring theoretic perspective. This will be discussed in terms of one particular example, quantum matrices - more accurately, the quantized coordinate rings of matrix varieties. These algebras will be introduced, their somewhat strange-looking relations will be motivated, and various properties will be discussed. Some of the surprising parallels between these algebras and the corresponding classical coordinate rings will be presented.

UNIVERSITY OF CALIFORNIA
SANTA BARBARA, CA
goodearl at math dot ucsb dot edu

The ideal of phantom maps and purity

PEDRO ANTONIO GUIL ASENSIO, University of Murcia

We show that a morphism in a locally finitely presented Grothendieck category C is phantom if and only if it is Ext-orthogonal to the class of pure-injective objects. The ideal of morphisms that are orthogonal to phantom morphisms is also studied. When the category C has enough projectives, this ideal consists of all morphisms that factor through a pureinjective objects. The existence of pre-envelopes with respect to this ideal of the category is also proved. Finally, it is shown that the class of pureinjective objects is the closure under extensions of the class of pure-injective objects. We also outline how to extend these results to exact categories.

Joint work with X. Fu, I. Herzog and B. Torrecillas.

UNIVERSITY OF MURCIA
MURCIA, SPAIN
paguil at um dot es

Idempotent matrices, sequences of ideals and

countably generated projective modules

DOLORS HERBERA, Universitat Autonoma de Barcelona

For any countable, column finite, idempotent matrix $E$ with entries in a ring $R$ we will describe how to construct a couple of sequences of ideals of $R$ that give some information on the countable generated projective module determined by $E$ .

I want to explain that for some classes of rings (e.g. semilocal noetherian rings) this information, combined with some realization methods, is enough to determine all projective modules over them. For some other classes of rings (e.g. general semilocal rings) this seems to be the starting point of a, surprisingly rich, theory of countably generated projective modules.

The talk is based on joint work with P. Prihoda.

UNIVERSITAT AUTONOMA DE BARCELONA
BELLATERRA , BARCELONA, SPAIN
dolors at mat dot uab dot cat

Rings whose modules have maximal or minimal projectivity domain

CHRISTOPHER HOLSTON, Ohio University

Using the notion of relative projectivity, projective modules may be thought of as being those which are projective relative to all others. In contrast, a module M is said to be projectively poor if it is projective relative only to semisimple modules. We prove that all rings have projectively poor modules. In fact, every ring even has a semisimple projectively poor module.

We consider rings over which modules are either projective or projectively poor and call them rings without a p-middle class. As we analyze the structure of rings with no right p-middle class, among other results, we show that any such ring is the ring direct sum of a semisimple artinian ring and a ring K which is either zero or an indecomposable ring such that either (i) K is a semiprimary right SI-ring with non-zero radical, or (ii) K is a semiprimary ring with right $Soc(K ) = Z_r (K) = J(K) ̸= 0$ ,or (iii) K is a prime ring with right Soc(K) = 0, and either J(K) = 0 or K J(K) and J(K)K are infinitely generated, or (iv) K is a prime right SI-ring with infinitely generated right socle. For a partial converse, we also give examples of rings with no p-middle class of types (i), (ii) and (iii). (Joint works with Sergio R. Lopez-Permouth and Nil Orhan-Ertas, as well as Hai Q. Dinh and Dinh van Huynh.)

OHIO UNIVERSITY
ATHENS, OH
ch327505 at ohio dot edu

pt

Jumps in the finitistic dimensions of finite dimensional algebras

BIRGE HUISGEN-ZIMMERMANN, University of California

I will start with a brief review of pre-2000 results on the Finitistic Dimension Conjectures (which date back to 1960), and then discuss some recent progress. In particular, I will address jumps in the functions $n \mapsto findim_n \Lambda$ for $n$ a natural number or $\infty$ , where $\Lambda$ is a finite dimensional algebra and $findim_n \Lambda$ is the supremum of the finite projective dimensions attained on $n$ -generated left $\Lambda$ -modules.

UNIVERSITY OF CALIFORNIA
SANTA BARBARA, CA
birge at math dot ucsb dot edu

Hausdorff topology on infinite nilpotent rings

MARTIN JURÁŠ, Qatar University

We prove that every infinite nilpotent associative or Lie ring admits a non-discrete locally totally bounded Hausdorff ring topology. An example of a nilpotent ring which does not admit a non-discrete bounded ring topology is given. (This is a joint work with Mihail Ursul.)

QATAR UNIVERSITY
DOHA, QATAR
martinjuras at gmail dot com

Idempotents and clean elements in polynomial rings

PRAMOD KANWAR, Ohio University - Zanesville

An element $a$ of a ring $R$ is called clean if $a=e+u$ for some idempotent $% e$ and some unit $u$ in $R$ . We obtain idempotents in polynomial rings and related ring extensions and use this information to give clean elements in polynomial rings and other related ring extensions. Among other things it is shown that $R$ is abelian if and only if there exists a positive integer $n$ such that $R[x]$ does not contain idempotents which are polynomials of degree $n$ . Idempotents of a polynomial ring which are conjugate to idempotents in the base ring are also studied. It is shown that if $R$ is either an abelian ring or a 2-primal ring such that $R[x]$ is an $ID$ -ring then every idempotent $e\in M_{n}(R)[x]$ is conjugated to a diagonal matrix of the form $diag(e_{1},\ldots ,e_{n})\in M_{n}(R)$ , where $e_{i}$ 's denote idempotents in $R$ . (This is a joint work with André Leroy and Jerzy Matczuk.)

OHIO UNIVERSITY - ZANESVILLE
ZANESVILLE, OHIO
kanwar at ohio dot edu

On Dedekind Criterion and simple extensions of valuation rings

SUDESH KAUR KHANDUJA,

Indian Institute of Science Education and Research (IISER, Mohali)

Let $K=\mathbb{Q}(\theta)$ be an algebraic number field with $\theta$ in the ring $A_K$ of algebraic integers of $K$ and $f(x)$ be the minimal polynomial of $\theta$ over the field $\mathbb{Q}$ of rational numbers. Dedekind proved that if a rational prime $p$ does not divide $[A_K:\mathbb{Z}[\theta]]$ , then the determination of the prime ideal decomposition in $A_K$ of $p$ is related to the decomposition of the polynomial $\bar{f}(x)$ obtained by replacing each coefficient of $f(x)$ by its residue modulo $p$ . Dedekind also gave a simple criterion known as Dedekind Criterion to verify when $p$ does not divide $[A_K:\mathbb{Z}[\theta]].$ In 2005 we extended Dedekind Criterion to relative extensions of algebraic number fields. An analogue of Dedekind Criterion for finite extensions of valued fields of arbitrary rank has also been formulated and proved. Our method of proof has led to a set of necessary and sufficient conditions which ensure when the integral closure $S$ of a valuation ring $R_v$ having quotient field $K$ in a finite extension $L$ of $K$ is a simple ring extension of $R_v$ , i.e., $S=v[\theta]$ for some $\theta$ . As a corollary, the classical Theorem of Dedekind characterizing those rational primes $p$ which divide the index of an algebraic number field follows.

INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH (IISER, MOHALI)
S. A. S. NAGAR MOHALI, PUNJAB, INDIA
skhanduja at iisermohali dot ac dot in

Rings of idempotent stable range one

DINESH KHURANA, Department of Mathematics, Panjab University

An element $a$ of a ring $R$ is said to have idempotent stable range one if for any $b$ in $R$ , $aR + bR = R$ implies that $a + be$ is a unit for some idempotent $e$ in $R$ . Clearly, an element having idempotent stable range one is clean. We prove that in a ring with stable range one, every regular element has idempotent stable range one. In particular, it follows that in a unit-regular ring every element has idempotent stable range one.

This is a joint work with Zhou Wang, Jianlong Chen and Tsit-Yuen Lam.

DEPARTMENT OF MATHEMATICS, PANJAB UNIVERSITY
CHANDIGARH, INDIA
dkhurana at pu dot ac dot in

Rings whose every ideals is absolute

KOMPANTSEVA EKATERINA IGOREVNA,

Moscow State Pedagogical University

A ring, whose additive group is isomorphic to an Abelian group $G$ is called a ring on $G$ . An absolute ideal of an Abelian group $G$ is a subgroup of $G$ , which is an ideal in every ring on $G$ . An Abelian group is called $RAI$ -group if it admits a ring structure, in which every ideal is absolute. The problem of $RAI$ -group description was formulated in [1, problem 93].

In this work, a description of reduced algebraically compact abelian $RAI$ -groups will be given. All groups, considered in this work, are Abelian and throughout this work "group" will mean "Abelian group".

A reduced algebraically compact group $G$ can be uniquely represented as $G=\prod \limits _p G_p$ , where $G_p$ is a regular direct sum $\sum \limits ^\sim _{i \in I} \mathbb{Q}^*_pe_i$ of cyclic $p$ -adic modules. Moreover, $G_p=p \oplus C_p$ , where $A_p$ is a reduced torsion-free $p$ -adic algebraically compact group, and $C_p$ is an adjusted $p$ -adic algebraically compact group [2].

Theorem 1. Let $G_p$ be a reduced $p$ -adic algebraically compact group, whose torsion subgroup is unbounded. Let $G_p=p \oplus C_p$ , where $A_p=\sum \limits ^\sim _{i \in I_A} \mathbb{Q}^*_pe_i$ is a reduced torsion-free $p$ -adic algebraically compact group, and $C_p=\sum \limits ^\sim _{i \in I_C} \mathbb{Q}^*_pe_i$ is an adjusted $p$ -adic algebraically compact group. Let $I_k=i \in I_C \mid o(e_i)=k\}$ , $m_A=_A\vert,\; m_k=_k\vert\;\;(k \in \mathbb{N})$ . Then
1) if $A \ne 0$ then $G$ is a $RAI$ -group if and only if there exists a whole number $n$ such that $m_A \ge \prod \limits _{k=}^\infty m_k$ .
2) If $A =$ , then $G$ is an $RAI$ -group if and only if $\sum \limits ^\infty _{k=} m_k \ge \prod \limits ^\infty _{k=} m_k$ for all $n \in \mathbb{N}$ .

Theorem 2. Let $G$ be a reduced algebraically compact group, $G=\prod \limits _{p \in \mathbb{P}} G_p$ , where $G_p$ is a reduced $p$ -adic algebraically compact group, whose torsion subgroup is unbounded. Then $G$ is a $RAI$ -group if and only if $G_p$ is a $RAI$ -group for every $p$ .

References

[1] L.Fuchs, Infinite Abelian groups // Academic Press New York and London, 1973.

[2] Kompantseva E.I. Torsion-free rings // Journal of Mathematical Sciences (New York), 2010, 171:2, 213-247.

MOSCOW STATE PEDAGOGICAL UNIVERSITY
MOSCOW, RUSSIA
kompantseva at yandex dot ru

A concept unifying semiprimeness and reversibility of rings

TAI KEUN KWAK, Daejin University

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this paper, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.

DAEJIN UNIVERSITY
POCHEON,
tkkwak at daejin dot ac dot kr

Commutators and generalized commutators in matrix rings

T. Y. LAM, University of California, Berkeley

An element of the form $[a,b]=ab-ba$ in a ring $R$ is called an (additive) commutator, and an element of the form $[a,b,c]=abc-cba$ in $R$ is called a generalized commutator. In this talk, we present some recent results (and problems) on commutators and generalized commutators in a matrix ring $R={\mathbb{M}}_n(S)$ over a (not necessarily commutative) base ring $S$ . This is joint work with D. Khurana and V. Kodiyalam.

UNIVERSITY OF CALIFORNIA, BERKELEY
BERKELEY, CALIFORNIA
lam at math dot berkeley dot edu

Modules whose endomorphism rings are von Neumann regular

GANGYONG LEE, The Ohio State University

The study of the class of von Neumann regular rings has been a topic of wide interest. In 1958 Fuchs raised the question of characterizing abelian groups whose endomorphism rings are von Neumann regular. This was answered for the case of groups by Rangaswamy in 1967 but independently in 1960 for the case of modules by Azumaya. In particular, it was shown that for a module $M$ , End$_R(M)$ is a von Neumann regular ring iff $Ker \varphi$ and $Im\varphi$ are direct summands of $M$ for every $\varphi\in$   End$_R(M)$ .

In this work we study modules $M$ which satisfy these two conditions namely, for all $\varphi\in$   End$_R(M)$ , $Ker \varphi$ is a direct summand of $M$ and $Im\varphi$ is a direct summand of $M$ . (In fact from our recent works, a module satisfying only the first condition is called Rickart and a module satisfying only the second one is called dual Rickart). In view of Azumaya and Rangaswamy results, we call a module $M$ satisfying the two conditions endoregular. We will present some new results on endoregular modules and other related notions in this talk. In particular, we will provide characterizations of modules whose endomorphism rings are: strongly regular, unit regular, division rings, simple artinian rings, and semisimple artinian rings, among other classes. Examples which delineate the concepts and results will be shown.

(This is a joint work with S. Tariq Rizvi and Cosmin Roman.)

THE OHIO STATE UNIVERSITY
COLUMBUS, OH
lgy999 at math dot osu dot edu

Intrinsic extensions of rings

MATTHEW (JAKE) LENNON, University of Louisiana Lafayette

In this talk the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, $R$ , a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, $T$ , from a corresponding set in the base ring $R$ . Examples and applications are given for rings that occur in Functional Analysis. Furthermore, it is shown that our main results provide a new method for attempting to investigate the well-known Zero Divisor Problem in Group Ring Theory and possibly extend partial solutions to the problem.

UNIVERSITY OF LOUISIANA LAFAYETTE
LAFAYETTE, LOUISIANA
jake.lennon at gmail dot com

Factorizations in Ore extensions

ANDRÉ GÉRARD LEROY, Université d'Artois

We will survey results on Ore polynomial rings obtained essentially in the 90's with T.Y.Lam. We will also present some more recent results on this topic. The focus will be on factorizations problems of skew polynomials mainly with coefficients in division rings. The Wedderburn polynomials, their different characterizations and properties their relations with Vandermonde and Wronskian matrices will be at the center of the talk. We will also consider the more general completely reducible polynomials. Using pseudo-linear transformation, we will indicate how some of the results can be naturally interpreted and generalized to the case when the base ring is not a division ring. Applications to some topics such as noncommutative symmetric functions and coding theory will be mentioned.

UNIVERSITÉ D'ARTOIS
LENS, PAS DE CALAIS, FRANCE
andreleroy55 at gmail dot com

Zero-divisor graphs of group rings

YUANLIN LI, Brock University

Let $R$ be a commutative ring with $1\neq 0$ , $G$ be a nontrivial finite group and let $Z(R)$ be the set of zero-divisors of $R$ . The zero-divisor graph of $R$ is defined as the graph ${\Gamma}(R)$ with the vertex set $Z(R)^*=R)\setminus\{(0)\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab=$ . In this talk, we investigate the interplay between the ring-theoretic properties of group ring $RG$ and the graph-theoretic properties of $\Gamma(RG)$ . We first characterize finite abelian group algebras $KG$ with $diam(\Gamma(KG))\leq 2$ as well as Artinian commutative group rings $RG$ with $gr(\Gamma(RG)\geq 4$ . We also investigate the isomorphic problem for zero-divisor graphs of group rings. It is shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. It is also shown that rank and cardinality of a finite abelian p-group is completely determined by the zero-divisor graph of its modular group ring, extending a result of Akbari et al (J. Algebra 2004). Finally, we show that a finite noncommutative reversible group ring is completely determined by its zero-divisor graph.

Joint work with Farid Aliniaeifard.

BROCK UNIVERSITY
ST. CATHARINES, ONTARIO, CANADA
yli at brocku dot ca

C.P. modules and their applications

QIONGLING LIU, Southeast University

Let $R$ be a ring. A left $R$ -module $M$ is called a c.p. module if every cyclic submodule of $M$ is projective. This notion is a generalization of left p.p. rings in the module theoretic setting. In this talk, some characterizations and properties of c.p. modules are discussed. As applications, the connections among Baer rings, p.p. rings and von Neumann regular rings are studied. For example, it is proved that the class of c.p. left $R$ -modules is closed under direct sums. We also show that $R$ is Baer if and only if $R$ is p.p. and the class of c.p. left $R$ -modules is closed under direct products. Moreover, von Neumann regular rings are characterized in terms of c.p. modules (joint with Professor Jianlong Chen).

SOUTHEAST UNIVERSITY
NANJING, CHINA
ahlql123 at 163 dot com

Several orthogonal classes of flat and FP-injective functors

LIXIN MAO, Institute of Mathematics, Nanjing Institute of Technology

In this talk, we develop some relative homological algebra in the category of functors from finitely presented modules to abelian groups. More specifically, we introduce the concepts of F-injective, F-projective and F-flat functors. These functors are discovered when we study covers and envelopes of functors. The relationships among these functors are investigated and some applications are given.

INSTITUTE OF MATHEMATICS, NANJING INSTITUTE OF TECHNOLOGY
NANJING, CHINA
maolx2 at hotmail dot com

An alternative perspective on projectivity of modules-

preliminary report

JOSEPH LAWRENCE MASTROMATTEO, Ohio University

Similar to the idea of relative projectivity, we introduce the notion of relative subprojectivity, which is an alternative way to measure the projectivity of a module. Given modules $M$ and $N$ , $M$ is said to be $N$ -subprojective if for every epimorphism $g:B \rightarrow N$ and homomorphism $f:M \rightarrow N$ , then there exists a homomorphism $h:M \rightarrow B$ such that $gh=f$ . For a module $M$ , the subprojectivity domain of $M$ is defined to be the collection of all modules $N$ such that $M$ is $N$ -subprojective. A module is projective if and only if its subprojectivity domain consists of all modules. Opposite to this idea, a module $M$ is said to be $p$ -indigent if its subprojectivity domain is as small as conceivably possible, that is, consisting of exactly the projective modules. Properties of subprojectivity domains and $p$ -indigent modules are studied. In particular, the existence of a $p$ -indigent module is attained for artinian serial ring.

This is joint work with Holston, Lopez-Permouth and Simental.

OHIO UNIVERSITY
ATHENS, OHIO
jm424809 at ohio dot edu

Commutator rings and Leavitt path algebras

ZACHARY MESYAN, University of Colorado, Colorado Springs

An associative ring $R$ is said to be a commutator ring if $R=[R,R]$ , where $[R,R]$ is the subgroup of $R$ generated by its additive commutators. There has been interest in such rings since at least 1956, when Kaplansky asked whether there could be a commutator division ring. To date, few examples of rings with this property have been produced, but it turns out that many such examples can be built using Leavitt path algebras. Commutator Leavitt path algebras have the additional unusual property that all their Lie ideals are (ring-theoretic) ideals. It is also possible to completely classify the commutator Leavitt path algebras, and in the process to describe the commutator subspace $[L_K(E), L_K(E)]$ of the Leavitt path algebra $L_K(E)$ , for any field $K$ and directed graph $E$ .

UNIVERSITY OF COLORADO, COLORADO SPRINGS
COLORADO SPRINGS, CO
zmesyan at uccs dot edu

On pseudo semisimple rings

HATICE MUTLU, Izmir Institute of Technology

A necessary and sufficient condition is obtained for a right pseudo semisipmle ring to be left pseudo semisimple. It is also proved that a right and left pseudo semisimple ring is an internal exchange ring and is an SSP ring.

IZMIR INSTITUTE OF TECHNOLOGY
IZMIR, TURKEY
hatcemutlu at gmail dot com

On a theorem of Camillo and Yu

WILLIAM KEITH NICHOLSON, University of Calgary

In 1995 Camillo and Yu showed that an exchange ring has stable range 1 if and only if every regular element is unit-regular. A definition of a stable module is given and the analogous theorem is proved.

UNIVERSITY OF CALGARY
CALGARY, ALBERTA
wknichol at ucalgary dot ca

The bounded nilradical

PACE P. NIELSEN, Brigham Young University

The set $B(R)=\{a\ :\ Ra$    is nil of bounded index$\}$ is an ideal. This was first proved by Amitsur, and then given another proof by Klein. Klein also proved that $B(R[x])=R)[x]$ . We provide new intuitive proofs for these facts, and show that these proofs generalize to skew polynomial rings more easily than the earlier proofs.

BRIGHAM YOUNG UNIVERSITY
PROVO, UT
pace at math dot byu dot edu

Polycyclic codes over Galois rings

HAKAN OZADAM, Ohio University, Department of Mathematics

We study polycyclic codes over Galois rings which are generalizations of cyclic codes. These codes have a nice algebraic structure as they can be viewed as ideals of a factor ring of a polynomial ring. We study the structure of the ambient ring of polycyclic codes. We show the existence of a certain type of a generating set for an ideal of this ring. This generating set turns out to be a strong Groebner basis. We provide a method for finding such sets over a Galois ring of characteristic $p^2$ . We also give a method to determine the Hamming distance of the constacyclic codes of length $np^s$ and $2np^s$ over a Galois ring of characteristic $p^a$ .

This is a joint work with Sergio R. Lopez-Permouth, Ferruh Ozbudak and Steve Szabo.

OHIO UNIVERSITY, DEPARTMENT OF MATHEMATICS, PHD STUDENT
ATHENS, OHIO
ho379511 at ohio dot edu

A class of *-clean rings

AYSE CIGDEM OZCAN, Hacettepe University

In this talk, some $*$ -clean rings will be introduced and various properties of them will be presented.

Joint work with H. Chen and A. Harmanci

HACETTEPE UNIVERSITY
ANKARA, TURKEY
ozcan at hacettepe dot edu dot tr

When Leavitt path algebras have bases

consisting entirely of units - Preliminary Report.

NICHOLAS J. PILEWSKI, Ohio University

Following Lopez-Permouth, Moore and Szabo, a $K$ -algebra $A$ is called an invertible algebra if it has a $K$ -basis of units in A. The purpose of this project is to determine the directed graphs $E$ for which the Leavitt path algebra $L_K(E)$ is an invertible algebra. Among other results, we show that all noetherian Leavitt path algebras are invertible. We also describe another family of invertible Leavitt path algebras which includes, for all $n,$ the classical Leavitt algebra $L(1, n).$ (This is joint work with Al-Essa, López-Permouth and Simental).

OHIO UNIVERSITY
ATHENS, OH
np338697 at ohio dot edu

Good gradings of generalized incidence rings

KENNETH PRICE, University of Wisconsin, Oshkosh

This inquiry is based on both the construction of generalized incidence rings due to Gene Abrams and the construction of good group gradings of incidence algebras due to Molli Jones. The speaker will provide conditions for a generalized incidence ring to be graded isomorphic to a subring of an incidence ring over a preorder. An extension of Jones’s construction to good group gradings for incidence algebras over preorders with crosscuts of length one or two will also be covered.

UNIVERSITY OF WISCONSIN, OSHKOSH
OSHKOSH, WI
pricek at uwosh dot edu

Lie ideals and Jordan triple derivations in rings

MURTAZA ALI QUADRI, Aligarh Muslim University

Let $R$ be an associative ring. An additive subgroup $L$ of $R$ is said to be a Lie ideal of $R$ if $[L, R] \subseteq L$ . An additive mapping $\delta : R \longrightarrow R$ is called a derivation (resp. a Jordan derivation) if $\delta(xy) =\delta(x)y + x\delta(y)$ (resp. $\delta(x^2) =\delta(x)x + x\delta(x))$ holds for all $x,y \in R$ . A famous result due to Herstein states that a Jordan derivation in a prime ring of characteristic not equal to $2$ must be a derivation. This result was extended to 2-torsion free semiprime rings by Cusack and subsequently, by Bresar.

Following Bresar [Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218-228], an additive mapping $d : R \longrightarrow R$ is called a Jordan triple derivation if $d(xyx) =d(x)yx + xd(y)x+xyd(x)$ holds for all $x,y \in R$ . One can easily prove that any Jordan derivation of a $2$ -torsion free ring is a Jordan triple derivation. An additive mapping $F : R\longrightarrow R$ is said to be a generalized Jordan triple derivation on $R$ if there exists a Jordan triple derivation $d : R \longrightarrow R$ such that $F(xyx)=d(x)yx + xd(y)x+xyd(x)$ holds for all $x,y \in R$ .

It is obvious to see that every derivation is a Jordan triple derivation but the converse need not be true in general. In 1989, Bresar proved that any Jordan triple derivation on a $2$ -torsion free semiprime ring is a derivation. In the present talk, my attempt will be to extend this study further and obtain some results on Lie idesls. Also two conjuctures will be presented for open discussion (if not settled by the time, the talk will be given).

DEPARTMENT OF MATHEMATICS, ALIGARH MUSLIM UNIVERSITY
ALIGARH, UTTER PRADESH, INDIA
maquadri1 at rediffmail dot com

Structure of certain conditioned rings and near rings

REKHA RANI, N. R. E. C. College

Using commutativity of rings satisfying $(xy)^{n(x,y)} =$ proved by Searcoid and MacHale [Amer. Math. Monthly 93 (1986)], Ligh and Luh [Amer. Math. Monthly 96 (1989)] gave a direct sum decomposition for rings with the
mentioned condition. Further, Bell and Ligh [Math. J. Okayama Univ.31 (1989)] sharpened the result and obtained a decomposition theorem for rings with the property $xy =y)^{2} f(x,y)$ where $f(X,Y) \in \mathbb{Z}<X,Y>$ , the ring of polynomials in two noncommuting indeterminates over the ring $\mathbb{Z}$ of integers. In the present paper, we continue the study and investigate structure of certain rings and near rings satisfying the following condition: $xy =p(x,y)$ with $p(x,y)$ , an admissible polynomial in $\mathbb{Z} <X,Y>$ . This condition is naturally more general than the above mentioned conditions. Moreover, we deduce the commutativity of such rings.

Theorem 1. Let  $R$   be a  ring such that for each $x,y \in R$ there exists an admissible polynomial $p(X,Y) \in \mathbb{Z}<X,Y>$ for which $xy =p(x,y)$ . Then $R$ is a direct sum of a $J$ -ring and a zero ring.

Theorem 2. Let $R$ be a d-g near ring such that for each $x,y \in R$ there exists an admissible polynomial $p(X,Y) \in \mathbb{Z}<X,Y>$ for which $xy =p(x,y)$ . Then $R$ is periodic and commutative. Moreover, $R =uplus N$ , where P, the set of potent elements of $R$ is a subring and N, the set of nilpotent elements of $R$ is a subnear ring with trivial multiplication.

N. R. E. C. COLLEGE
BULANDSHAHAR, UTTER PRADESH, INDIA
rekharani at rediffmail dot com

Obstructing extensions of the functor Spec to noncommutative rings

MANUEL LIONEL REYES, Bowdoin College

I will present the following obstruction result for functors extending the Zariski spectrum to noncommutative rings: every contravariant functor from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to $\mathrm{Spec}$ must assign the empty set to $\mathbb{M}_n(\mathbb{C})$ for $n \geq 3$ . The proof relies on the Kochen-Specker ``no-hidden-variables'' theorem of quantum mechanics. I will also mention a recent generalization of the result due to van den Berg and Heunen.

BOWDOIN COLLEGE
BRUNSWICK, ME
reyes at bowdoin dot edu

"Klein's idea and identities for powers of polynomials"

BRUCE REZNICK, University of Illinois

In his famous book on the icosahedron, Felix Klein considered the identification of a point on the unit sphere u, with the linear form x - vy, where v is the image of u under the Riemann map. If you start with the vertices of a nice polytope inscribed in the sphere, and take quadratic forms corresponding to products of linear forms associated with antipodal pairs of vertices, you get interesting sets of quadratic forms. For example, the octahedron corresponds to $\{xy, x^2-y^2,x^2+y^2\}$ (the Pythagorean parameterization), the cube to four quadratic forms whose 5-th powers are dependent and the icosahedron to six quadratic forms whose 14-th powers are dependent. We'll give more examples and try to explain this phenomenon.

UNIVERSITY OF ILLINOIS
URBANA, IL
reznick at math dot uiuc dot edu

Endo-Rickart modules

COSMIN ROMAN, The Ohio State University, Lima

A left Rickart (respectively, left Baer) ring is one in which the left annihilator of any element (respectively, arbitrary nonempty subset) is generated by an idempotent. Right-sided notions are defined similarly. It is well-known that the notion of a Baer ring is always left-right symmetric while the notion of a Rickart ring is not left-right symmetric.

We recently extended the right Rickart property of rings to a module-theoretic setting, and called it a Rickart module. On the other hand, the study of left Rickart rings in this general setting remains open. In this talk we present a module theoretic analogue of a left Rickart ring and call it an Endo-Rickart module. A $R$ -module $M$ is called Endo-Rickart if the left annihilator in $S$ of an element of $M$ is a left direct summand in $S$ .

In this talk we present results and properties of Endo-Rickart modules and ancillary notions. Examples and applications will also be provided.

(This is a joint work with G. Lee and S. Tariq Rizvi.)

THE OHIO STATE UNIVERSITY, LIMA
LIMA, OH
cosmin at math dot osu dot edu

Mittag-Leffler modules

PHILIPP ROTHMALER, Graduate Center of CUNY

The Mittag-Leffler property constitutes a useful generalization of (pure) projectivity. I will discuss some new results about it.

GRADUATE CENTER OF CUNY
NEW YORK, NY
philipp.rothmaler at bcc dot cuny dot edu

Fully invariant decompositions of modules using

pairwise comaximal ideals

CHRISTOPHER EDWARD RYAN, University of Louisiana at Lafayette

We introduce a general decomposition for a large class of modules over rings with unity. In particular, it is shown that for a given set of pairwise comaximal ideals $\{X_1,...,X_n\}$ in the ring, if $M$ is a right $R$ -module such that $\bigcap\limits_{i=}^nX_i\subseteq\underline{r}_R(M)$ , then $M$ is a direct sum of the $\underline{\ell}_M(X_i)$ , where $\underline{\ell}_M(X_i)=\{m\in M\vert mX_i=0\}$ . Several well- known results from commutative ring theory and the theory of semisimple Artinian rings are presented as examples of our main theorem. Also some torsion theoretical results related to our decompositions will be shown. This talk is based on joint work with Gary F. Birkenmeier.

UNIVERSITY OF LOUISIANA AT LAFAYETTE
LAFAYETTE, LOUISIANA
cxr2665 at louisiana dot edu

Operations on arc diagrams and degenerations

for invariant subspaces of linear operators

MARKUS SCHMIDMEIER, Florida Atlantic University

We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are reductive algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described. This is a report about a joint project with Justyna Kosakowska from Torun in Poland.

FLORIDA ATLANTIC UNIVERSITY
BOCA RATON,
markus at math dot fau dot edu

Primitive idempotents in Leavitt path algebras

MERCEDES SILES MOLINA, Universidad de Malaga (Spain)

We will start this talk by showing the current picture concerning the classification of Leavitt path algebras. We will see that primitive idempotents play an essential role, in particular, the primitive ones. We will introduce the notion of truly primitive idempotent, closely related to Condition (L) and will show its implication in such a classification.

UNIVERSIDAD DE MALAGA
MALAGA, SPAIN
msilesm at uma dot es

Studying rings in terms of the extent of injectivity

and projectivity of their modules

JOSE EDUARDO SIMENTAL, Ohio University

Given a ring $R$ , we define its right $i$ -profile (resp. right $p$ -profile) as the collection of injectivity domains (resp. projectivity domains) of its right modules. We show that the $i$ -profile of a ring has a natural lattice structure, and we study its properties, while the $p$ -profile has a semilattice structure. We also characterize those rings for which the projectivity and injectivity domain of every module coincide. The study of the profile(s) of a ring has various connections to torsion theory and with the theory of poor modules and rings with no middle class, recently introduced by Er, López-Permouth, Sökmez, Holston and Orhan-Ertas. (Joint work with Sergio R. López-Permouth)

OHIO UNIVERSITY
ATHENS, OHIO
jesr_ at hotmail dot com

Dual automorphism-invariant modules

ASHISH K. SRIVASTAVA, St. Louis University

We call a module $M$ to be a dual automorphism-invariant module if whenever $K_1$ and $K_2$ are small submodules of $M$ , then any epimorphism $\eta:M/K_1\rightarrow M/K_2$ with small kernel lifts to an endomorphism $\varphi$ of $M$ . In this talk we will give various examples of dual automorphism-invariant modules and discuss their properties. (This is a joint work with Surjeet Singh)

ST. LOUIS UNIVERSITY
SAINT LOUIS, MO
asrivas3 at slu dot edu

The Morita invariance problem for clean rings

JANEZ ŠTER, Institute of Mathematics, Physics and Mechanics

An element of a ring $R$ is called clean if it can be written as a sum of an idempotent and a unit in $R$ . A ring is called clean if its every element is clean. Clean rings arise as a special example of exchange rings. An example due to Bergman shows that an exchange ring does not need to be clean.

It is known that the clean property is closed under matrix extensions, meaning that if $R$ is clean, then $M_n(R)$ is clean. The converse of this proposition is still an open problem. More generally, it is not known whether $R$ being clean implies that $eRe$ is clean for a full idempotent $e\in R$ .

In the talk, we classify all rings $R$ with the property that the matrices $\left(\begin{smallmatrix}a&0\\ 0&0\end{smallmatrix}\right)$ are clean in $M_2(R)$ for every $a\in R$ . We show that the Bergman's ring satisfies this property, and using this observation, we give an example of a clean ring $R$ and an idempotent $e\in R$ such that the corner ring $eRe$ is not clean.

INSTITUTE OF MATHEMATICS, PHYSICS AND MECHANICS
LJUBLJANA, SLOVENIA
janez.ster at fmf dot uni-lj dot si

Preserving zeros of $xy$ and $xy^\ast$

NIK STOPAR, University of Ljubljana

We describe surjective additive maps $\theta : A \to B$ which preserve zero products (i.e. $xy=0$ implies $\theta(x)\theta(y)=0$ ), where $A$ is a ring with a nontrivial idempotent and $B$ is a prime ring. We also characterize surjective additive maps $\theta : A \to B$ such that $\theta(x)\theta(y)^\ast=0$ if and only if $xy^\ast=0$ , where $A$ is a unital prime ring with involution that contains a nontrivial idempotent and $B$ is a prime ring with involution.

UNIVERSITY OF LJUBLJANA
LJUBLJANA, SLOVENIA
nik.stopar at fmf dot uni-lj dot si

On dual Baer modules

SULTAN EYLEM TOKSOY, Izmir Institute of Technology

In this note we prove that any ring $R$ is right cosemihereditary if and only if every finitely cogenerated injective right $R$ -module is $d$ -Rickart. We also prove that if $M=M_1\oplus M_2$ with $M_2$ semisimple, then $M$ is dual Baer if and only if $M_1$ is dual Baer and every simple non-direct summand of $M_1$ does not embed in $M_2$ .

Joint work with Derya Keskin Tütüncü and Patrick F. Smith.

IZMIR INSTITUTE OF TECHNOLOGY
IZMIR, TURKEY
eylemtoksoy at iyte dot edu dot tr

Flat Mittag-Leffler modules

JAN TRLIFAJ, Univerzita Karlova

We present some recent results from [1]-[6] on flat Mittag-Leffler ($=\aleph_1$ -projective) modules, and their consequences for generalized vector bundles.

References.

S. BAZZONI, J. ŠŠTOV´IŠCEK, Flat Mittag-Leffler modules over countable rings, Proc. Amer. Math. Soc. 140 (2012), 1527-1533.

G. BRAUN, J. TRLIFAJ, Strong submodules of almost projective modules, Pacific J. Math. 254 (2011), 73-87.

S. ESTRADA, P. GUIL ASENSIO, J. TRLIFAJ, Descent of restricted flat Mittag-Leffler modules and generalized vector bundles, arXiv:1110.5364.

Infinite-dimensional vector bundles in algebraic geometry: an introduction, in 'The Unity of Mathematics', Birkhäuser, Boston 2006, 263-304.

D.HERBERA, J.TRLIFAJ, Almost free modules and Mittag-Leffler conditions, Advances in Math. 229(2012), 3436-3467.

J.ŠAROCH, J.TRLIFAJ, Kaplansky classes, finite character, and $\aleph_1$ -projectivity, Forum Math. 24(2012).

UNIVERZITA KARLOVA
PRAGUE, CZECH REPUBLIC
trlifaj at karlin dot mff dot cuni dot cz

On a class of $\oplus$ -supplemented modules

BURCU UNGOR, Ankara University

We introduce principally $\oplus$ -supplemented modules as a generalization of $\oplus$ -supplemented modules and principally lifting modules. This class of modules is a strengthening of principally supplemented modules. We show that the class of principally $\oplus$ -supplemented modules lies strictly between classes of $\oplus$ -supplemented modules and principally supplemented modules. We prove that some results of $\oplus$ -supplemented modules and principally lifting modules can be extended to principally $\oplus$ -supplemented modules for this general settings. We obtain some characterizations of principally semiperfect rings and von Neumann regular rings by using principally $\oplus$ -supplemented modules.

ANKARA UNIVERSITY
ANKARA, TURKEY
burcuungor at gmail dot com

Three ways in which T. Y. Lam impacted my life

LIA VAS, University of the Sciences

The talk presents three directions of research motivated by T. Y. Lam's work. The inspiration comes from: (1) Lam's view of classical rings of quotients as ``the Good, the Bad and the Ugly'' from ``Lectures on Rings and Modules''; (2) Lam's question ``Which von Neumann algebras are clean as rings?'' from the Athens, OH conference in 2005; (3) Lam's treatment of uniform dimension in ``Lectures on Rings and Modules''.

UNIVERSITY OF THE SCIENCES
PHILADELPHIA, PA
l.vas at usciences dot edu

D3-modules and D3-covers

MOHAMED F. YOUSIF, The Ohio State University at Lima

A right $R$ -module $M$ is called a $D3$ -module, if $% M_{1}$ and $M_{2}$ are direct summands of $M$ , with $M=M_{1}+M_{2},$ then $% M_{1}\cap M_{2}$ is a direct summand of $M$ . Besides projective, quasi-projective and direct-projective modules, examples of $D3$ -modules include discrete and quasi-discrete modules, uniform and indecomposable modules, semisimple modules, Baer modules and $SIP$ -modules. In this talk we obtain several interesting and new characterizations of several well-known classes of rings in terms of $D3$ -modules. For example, we will show that a ring $R$ is right perfect if and only if every flat right $R$ -module is a $D3$ -module, and $R$ is right (semi)hereditary if and only if every principal right ideal of $S=End(F_{R})$ is a $D3$ -module, for any (finitely generated) free right $R$ -module $F.$ Following H. Bass, an $R$ -homomorphism $\phi :P\rightarrow M$ is called a $D3$ -cover of the right $R$ -module $M$ , if $P$ is a $D3$ -module, $\phi$ is an epimorphism, and $\ker \phi$ is small in $P.$ We will show that a ring $R$ is right (semi)perfect if and only if every (finitely-generated) right $R$ -module has a $D3$ -cover, and a ring $R$ is semiregular if and only if every finitely presented right $R$ -module has a $D3$ -cover. While projectivity is viewed more generally as a homological condition, the strength and generality of our results stems from the fact that the $D3$ -condition is a latticial condition and the proofs are module theoretic proofs. At the end of our talk we will provide a dualization of these results to $C3$ -modules.

This is a joint work with Professor Ismail Amin of Cairo University and our Ph.D. student Mr. Yasser Ibrahim of Cairo University.

THE OHIO STATE UNIVERSITY AT LIMA
LIMA, OHIO
yousif.1 at osu dot edu

On the existence of nonzero injective covers

and projective envelopes of modules

XIAOXIANG ZHANG, Southeast University

In general, the injective cover (projective envelope) of a simple module can be zero. A ring $R$ is called a weakly left V-ring (strongly left Kasch ring) if every simple left $R$ -module has a nonzero injective cover (projective envelope). It is proved that every nonzero left $R$ -module has a nonzero injective cover if and only if $R$ is a left artinian weakly left V-ring. Dually, every nonzero left $R$ -module has a nonzero projective envelope if and only if $R$ is a left perfect right coherent strongly left Kasch ring. Some related rings and examples are concerned. (This is a joint work with X. Song)

DEPARTMENT OF MATHEMATICS, SOUTHEAST UNIVERSITY
NANJING, CHINA
z990303 at seu dot edu dot cn

Rings of small clean index

YIQIANG ZHOU, Memorial University of Newfoundland

Motivated by recent work on uniquely clean rings, we introduce and discuss the clean index of a ring. Rings of small clean index are characterized. (Joint work with Tsiu-Kwen Lee)

MEMORIAL UNIVERSITY OF NEWFOUNDLAND
ST.JOHN'S, CANADA
zhou at mun dot ca

Some results on McCoy and Armendariz rings

MICHAL ZIEMBOWSKI, Warsaw University of Technology

A ring $R$ is called right McCoy if whenever non-zero polynomials $f(x) =0 + a_1x +\dots + a_mx^m$ and $g(x) =0 + b_1x +\dots + b_nx^n$ in $R[x]$ satisfy $f(x)g(x) =$ , then $f(x)r =$ for some $r\in R$ . A ring $R$ is Armendariz if $f(x)g(x) =$ implies $a_ib_j =$ for all $i, j$ and $f(x) =um_{i=^na_ix^i}, \, g(x) =um_{j=^mb_jx^j} \in R[x]$ . In my talk I am going to present some new results concerning above mentioned classes of rings. (This talk is based on joint work with Ryszard Mazurek.)

WARSAW UNIVERSITY OF TECHNOLOGY
WARSAW, MAZOWIECKIE, POLAND
m.ziembowski at mini dot pw dot edu dot pl