Ring Theory Abstracts

Some Characterizations of $\delta$ -semiperfect and $\delta$ -perfect Rings.

PINAR AYDOGDU, Hacettepe University

Center of Ring Theory and its Applications, Ohio University

In this talk, we will present some characterizations of $\delta$ -semiperfect and $\delta$ -perfect rings in terms of locally (finitely, quasi-, direct-) projective $\delta$ -covers and flat $\delta$ -covers. Also, we investigate some properties of flat modules having a projective $\delta$ -cover.

HACETTEPE UNIVERSITY, ANKARA, TURKEY;
CENTER OF RING THEORY AND ITS APPLICATIONS, OHIO UNIVERSITY,
ATHENS, OHIO
paydogdu at hacettepe dot edu dot tr

Extensions of Commutative Rings

PAPIYA BHATTACHARJEE, Penn State Erie - The Behrend College

In studying the minimal prime spectra of commutative rings with identity, we have been able to identify several interesting types of extensions of rings, namely, $m$ -extension, rigid extensions, $r$ -extension, and $r^*$ -extension. In this talk, we will introduce these extensions for commutative rings and establish some relationship between them. In particular, we will determine what kind of ring extensions will result in a homeomorphism of the topologies of the minimal prime spectrum.

PENN STATE ERIE - THE BEHREND COLLEGE
ERIE, PA
pxb39 at psu dot edu

30 years of clean rings in 20 minutes

VICTOR PETER CAMILLO, University of Iowa

It all starts with Fitting's Lemma. Some roads it follows.

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

Linear Kerdock codes and relative difference sets in Galois rings

YUQING CHEN, Wright State University

In this talk I will explain the equivalence between $\mathbb{Z}_4$ -linear Kerdock codes and certain relative difference sets in Galois rings over $\mathbb{Z}_4$ .

WRIGHT STATE UNIVERSITY
DAYTON, OH
yuqing dot chen at wright dot edu

Associated classes of modules

IULIU CRIVEI, Babes-Bolyai University

Let $\mathcal{C}$ be a non-empty class of modules closed under isomorphic copies. We consider some classes of modules associated to $\mathcal{C}$ . Among them, we study two important classes in the theory of natural and conatural classes of modules, namely the class consisting of all modules having no non-zero submodule in $\mathcal{C}$ , as well as its dual. (joint work with Septimiu Crivei)

BABES-BOLYAI UNIVERSITY
CLUJ-NAPOCA, ROMANIA
crivei at math dot utcluj dot ro

Coslenderness revisited

RADOSLAV DIMITRIC

truecm

I will look into notions that are in some sense dual to the notion of slenderness. I will also address some fallacies regarding some of the dual notions that have persisted for many years.

PITTSBURGH, PA
rdimitric at juno dot com

Structure and distances of some classes of

constacyclic codes over Galois rings

HAI DINH, Kent State University

We discuss the structure of negacyclic codes of length $2^s$ over the Galois rings $GR(2^a,m)$ . It will be shown that the ambient ring $\frac{GR(2^a,m)[x]}{\langle x^{2^s}+ \rangle}$ is a chain ring, whose ideals are such negacyclic codes. This structure are then used to completely determined the Hamming and homogeneous distances of all such codes. The technique is then generalized to obtain the structure and Hamming and homogeneous distances of all $\gamma$ -constacyclic codes of length $2^s$ over $GR(2^a,m)$ , where $\gamma$ is any unit of the ring $GR(2^a,m)$ that has the form $\gamma=(4k_0-1)+4k_1\xi+\cdots+4k_{m-1}\xi^{m-1}$ , for integers $k_0, k_1,\dots,k_{m-1}$ . Among other results, duals of such $\gamma$ -constacyclic codes are studied, and necessary and sufficient conditions for the existence of a self-dual $\gamma$ -constacyclic code are established.

KENT STATE UNIVERSITY
WARREN, OH
hdinh at kent dot edu

A result on WV-rings.

DINH VAN HUYNH, Department of Mathematics, Ohio University

A module $M$ is called a V-module if every simple module in the category $\sigma[M]$ is $M$ -injective. In their recent paper, Holston, Jain and Leroy introduced the concept of WV-rings: A ring $R$ is called a right WV-ring if every cyclic right $R$ -module is either isomorphic to $R_R$ or it is a $V$ -module. In this talk, we show that if $R$ is a right WV-ring which is not right V, then $R$ has exactly 3 distinct right ideals $0 \subset N \subset R$ such that $R/N$ is a division ring, and $N2 = 0$ .

This is a joint work with C. Holston.

DEPARTMENT OF MATHEMATICS, OHIO UNIVERSITY
ATHENS, OH
huynh at math dot ohiou dot edu

Zassenhaus Rings as Idealizations of Modules

MANFRED DUGAS, Baylor University

A ring $R$ is called a Zassenhaus ring if any homomorphism $\varphi$ of the additive group of $R$ that leaves all left ideals of $R$ invariant, is a left multiplication by some element a of $R$ , i dot e. $\varphi(x)=ax$ for all $x\in R$ . Let $M$ be a $R-R$ -bimodule. Then the direct sum $R\oplus M$ turns naturally into a ring $R(+)M$ by defining $MM=\{0\}$ . This ring is called the idealization of the module $M$ , which is an ideal of $R(+)M$ . We will investigate conditions under which $R(+)M$ is a Zassenhaus ring.

BAYLOR UNIVERSITY
WACO, TX
Manfred_Dugas at baylor dot edu

On the endofiniteness of a key module over pure semisimple 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. In 2007, L. Angeleri Hügel introduced and studied a key module $W$ in $R$ -mod, where $R$ is hereditary left pure semisimple and $W$ is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective modules in $R$ -mod. The module $W$ appears to play a significant role in understanding how far the left pure semisimple ring $R$ is from being of finite representation type. In this talk we provide an answer to a question by Angeleri Hügel, showing that if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in $R$ -mod, then the key module $W$ is endofinite if and only if the ring $R$ has finite representation type. (This is joint work with José Luis García from the University of Murcia, Spain)

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

The Jacobson Radical's Role in Isomorphism Theorems

MARY K. FLAGG, University of Houston

The subject of isomorphism theorems began with the Baer-Kaplansky Theorem, which states that two abelian groups are isomorphic if and only if there exists a ring isomorphism between their respective endomorphism rings. Subsequent research by Wolfson, May and others has shown that several classes of modules over a discrete valuation domain also possess similar isomorphism theorems. I will show that in many classes of modules over a complete discrete valuation domain, an algebra isomorphism between only the Jacobson radical of the endomorphism rings of two modules is sufficient to imply that the modules are isomorphic.

UNIVERSITY OF HOUSTON
HOUSTON, TX
mflagg at math dot uh dot edu

Weak-injective modules

LÁSZLÓ FUCHS, Tulane University

All modules are over an integral domain $R$ . An $R$ -module $M$ is said to be weak-injective (S.B. Lee) if $Ext^1(G,M)=0$ for all $R$ -modules of weak-dimension at most 1. (This generalizes Enochs-cotorsion modules which are defined in the same way by using flat modules $G$ .) We survey some of the recent results on weak-injective modules.

TULANE UNIVERSITY
NEW ORLEANS, LA
fuchs at tulane dot edu

Module theory for associative rings

JOSE L. GARCIA, University of Murcia

We show how to construct a theory of modules over general rings (i dot e., rings which do not necessarily have an identity element) so that most of the known basic results of the theory of modules for unital rings are obtained by particularizing the results of the general theory. (Joint work with L. Marin)

UNIVERSITY OF MURCIA
MURCIA, SPAIN
jlgarcia at um dot es

Rings whose cyclics are direct sums of projective and

CS or noetherian

CHRIS HOLSTON, Ohio University

This is a summary of results of a paper with Jain and Leroy. $R$ is called a right $WV$ -ring if each simple right $R$ -module is injective relative to proper cyclics. If $R$ is a right $WV$ -ring, then $R$ is right uniform or a right $V$ -ring. It is shown that for a right $WV$ -ring $R$ , $R$ is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS or noetherian module. For a finitely generated module $M$ with projective socle over a $V$ -ring $R$ such that every subfactor of $M$ is a direct sum of a projective module and a CS or noetherian module, we show $M=X\oplus T$ , where $X$ is semisimple and $T$ is noetherian with zero socle. In the case that $M=R$ , we get $R=S\oplus T$ , where $S$ is a semisimple artinian ring, and $T$ is a direct sum of right noetherian simple rings with zero socle. In addition, if $R$ is a von Neumann regular ring, then it is semisimple artinian.

DEPARTMENT OF MATHEMATICS, OHIO UNIVERSITY
ATHENS, OH
holston at math dot ohiou dot edu

Clean elements and ring extensions

PRAMOD KANWAR, Ohio University, Zanesville

It is known that the polynomial ring $R[x]$ is never clean. Among other things, clean elements of $R[x]$ and $R[x,x^{-1}]$ are described.

OHIO UNIVERSITY, ZANESVILLE
ZANESVILLE, OH
pkanwar at ohiou dot edu

On GP-injective essential maximal right ideals

JIN YONG KIM, Kyung Hee University

Throught this paper all rings considered are associative with identity and all modules are unitary. We investigate in this paper the von Neumann regularity of rings whose essential maximal right ideals are GP-injective. We prove that a ring R is strongly regular if and only if R is a 2-primal ring whose essential maximal right ideals are GP-injective. It is also shown that a ring R is strongly regular if and only if R is a strongly right bounded ring whose essential maximal right ideals are GP-injective. Moreover we show that if R is a PI-ring whose esstial maximal right ideals are GP-injective, then R is strongly phi-regular.(This is a joint work with N.K.Kim and S.B.Nam)

KYUNG HEE UNIVERSITY
YONGIN-SI, GYEONGGI-DO, SOUTH KOREA
jykim at khu dot ac dot kr

Unique Decomposition of Direct Sums of Ideals

LEE KLINGLER, Florida Atlantic University

One says that the Krull-Schmidt Theorem holds for a class of modules if each module in the class can be decomposed uniquely into a direct sum of indecomposable modules in the class (up to order and isomorphism class of the summands). For most rings, even restricted to the class of finitely generated modules, this fails. In joint work with Basak Ay, we examine this question for commutative rings R and the category of R-modules which are isomorphic to finite direct sums of ideals of R.

FLORIDA ATLANTIC UNIVERSITY
BOCA RATON, FL
klingler at fau dot edu

Invertible commutators in rings, and

determinantal formulas in linear algebra

T. Y. LAM, University of California

An equation of the form $xy-yx=u\in {\rm U}(R)$ in called a unit metro equation in a ring $R$ . Whenever such an equation holds, we say that $x$ is ``completable'', and $xy$ is ``reflectable''. The study of such elements constitutes an interesting part of the understanding of the noncommutative nature of the ring $R$ . In this talk, we'll focus on a prototypical case; namely, where $R$ is a $2\times 2$ matrix ring over a commutative ring $S$ . In this case, the completable and reflectable elements of $R$ are closely tied to the ring arithmetic of $S$ . This study has led to the discovery of some determinantal formulas for computing ${\rm det}\,(XY-YX)$ . A ``trace version'' and a ``supertrace version'' turn out to be special cases of a general ``quantum-trace determinantal formula'' for computing ${\rm det}\,(XY-YX)$ . (This is a joint work with D. Khurana and N. Shomron.)

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

The dual Rickart property for modules

GANGYONG LEE, The Ohio State University

Let $R$ be any ring with unity and $M$ be a unital right $R$ -module. Set $S =End_R(M)$ . In 2004, Rizvi and Roman introduced the notion of a Baer module using the endomorphism ring of the module. $M$ is said to be Baer if the right annihilator in $M$ of any subset of $S$ is generated by an idempotent in $S$ . Recently, we extended this notion to one that generalizes the notion of a Baer module as well as that of a p dot p.-ring: $M$ is said to be Rickart if the right annihilator in $M$ of any single element of $S$ is generated by an idempotent $e=e^2\in S$ .

In this talk, we introduce a notion dual to that of a Rickart module. Namely, $M$ is called a dual Rickart module if the image in $M$ of any single element of $S$ is generated by an idempotent of $S$ . We present results and properties of dual Rickart modules and related concepts. We explore conditions which allow for direct sums of dual Rickart modules to be dual Rickart. In particular, we will discuss characterizations of classes of rings using the dual Rickart property of modules over them.

(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

Pseudo-linear transformations and applications

ANDRÉ LEROY, Université d'Artois

We will show that pseudo-linear transformations naturally appear in the context of a module of an Ore extension and give some of their properties. We will then briefly describe their relations with polynomial maps and factorizations in Ore extensions In particular, we will show how to translate factorization of skew polynomials over a finite field twisted by the Frobenius automorphism in terms of factorization of the usual polynomial ring over a finite field.

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

Poor Modules: The Opposite of Injectivity

SERGIO ROBERTO LÓPEZ-PERMOUTH, Ohio University

A module M is said to be poor if it is N-injective only when the module N is semisimple. This talk reports on results about Poor Modules and related notions obtained in recent collaborations with Adel Alahmadi, Mustafa Alkan, Noyan Er, and Nurhan Sokmez.

Among other results, we will show that all rings have poor modules and will precisely characterize those rings that have semisimple poor modules. We also address the question: over which rings does each module have either maximal or minimal domain of injectivity (i dot e. when is each module either poor or injective)? We will present a theorem classifying those rings (to which we refer as "rings without a middle class") and offer various components of a potential converse.

Addressing these questions will highlight how the notion of poor modules and related concepts relate to other widely studied families of rings and modules such as SI-rings, PCI domains, QI rings, etc.

OHIO UNIVERSITY
ATHENS, OH
lopez at ohio dot edu

Algebras having bases consisting entirely of units

JEREMY MOORE, Ohio University

We introduce a hierarchy of notions about algebras having a basis $B$ consisting entirely of units. Such a basis is called an invertible basis and algebras that have invertible bases are said to be invertible algebras. The other conditions considered in the said hierarchy include the requirement that for an invertible basis $B$ , the set of inverses $B^{−1}$ be itself a basis, the notion that $B$ be closed under inverses and the idea that $B$ be closed under products. It is shown that the last property is unique of group rings. Many examples are considered and it is determined that the hierarchy is for the most part strict. For any field $F = F_2$ , all semisimple $F$ -algebras are invertible. Semisimple invertible $F_2$ -algebras are fully characterized. Likewise, the question of which single-variable polynomials over a field yield invertible quotient rings of the $F$ -algebra $F[x]$ is completely answered. Connections between invertible algebras and $S$ -rings (rings generated by units) are also explored.

OHIO UNIVERSITY
ATHENS, OH
moore at math dot ohiou dot edu

(*)-generalized projective modules and lifting modules

NIL ORHAN ERTAS, Karabuk University

Center of Ring Theory and Its Applications, Ohio University

A module $M$ is called a lifting module if, any submodule $A$ of $M$ > contains a direct summand $B$ of $M$ such that $A/B$ is small in $M/B$ . It is known that a finite direct sum of lifting modules need not be lifting. It was investigated this question by using (*)-generalized projective modules (joint work with Ummuhan Acar).

KARABUK UNIVERSITY
TURKEY
CENTER OF RING THEORY AND ITS APPLICATIONS, OHIO UNIVERSITY
ATHENS, OH
orhannil at yahoo dot com

Ring structures on injective hulls of Artin algebras

BARBARA L. OSOFSKY, Rutgers University - Emerita

This is currently in a preliminary state. I will send an abstract after additional work.

RUTGERS UNIVERSITY
HIGHLAND PARK, NJ
osofskyb at member dot ams dot org

The Factor Ring of a Quasi-Baer Ring by its Prime Radical

JAE KEOL PARK, Department of Mathematics, Busan National University

A ring $R$ is called quasi-Baer if the right annihilator of every ideal of $R$ is generated by an idempotent. The quasi-Baer condition of $R/P(R)$ is discussed when $R$ is a quasi-Baer ring, where $P(R)$ is the prime radical of $R$ . We provide an example of quasi-Baer ring $R$ such that $R/P(R)$ is not quasi-Baer. However, when $P(R)$ is nilpotent, it is shown that if $R$ is a quasi-Baer (resp., Baer) ring, then $R/P(R)$ is quasi-Baer (resp., Baer). Examples which illustrate and delimit the results of this paper are also discussed (this is a joint work with Gary F. Birkenmeier and Jin Yong Kim).

DEPARTMENT OF MATHEMATICS, BUSAN NATIONAL UNIVERSITY
BUSAN 609-735, SOUTH KOREA
jkpark at pusan dot ac dot kr

A (one-sided) Prime Ideal Principle for noncommutative rings

MANUEL LIONEL REYES, University of California, Berkeley

In earlier work with T.Y. Lam, we unified several results from commutative algebra of the form "maximal implies prime" through the "Prime Ideal Principle." After a brief review of the commutative case, we will discuss a generalization of this method to noncommutative rings. This requires us to define a new type of "prime right ideal." We will illustrate these new ideas with several examples and applications.

UNIVERSITY OF CALIFORNIA, BERKELEY
ALBANY, CA
mreyes at math dot berkeley dot edu

Rickart Property for Projective Modules

COSMIN ROMAN, The Ohio State University

Let $R$ be any ring and $M$ be an $R$ -module. Set $S =End_R(M)$ . In 2004, we introduced the notion of a Baer module, generalizing the ring theoretic concept of Baer rings. $M$ is said to be Baer if the right annihilator in $M$ of any subset of $S$ is generated by an idempotent in $S$ . Equivalently, the left annihilator in $S$ of any submodule of $M$ is generated by an idempotent in $S$ . We recently extended this notion to that of a Rickart module, bringing the ring theoretic concept of Rickart (or PP-) rings to module theoretic setting. $M$ is called a Rickart module if the right annihilator in $M$ of any single element of $S$ is generated by an idempotent in $S$ , equivalently, $r_M(\varphi)=Ker \varphi \le^\oplus M$ for every $\varphi \in S$ .

In this talk we present results and properties of Rickart modules and related concepts. It is known that the direct sum of Rickart modules is not Rickart in general. We explore conditions which allow for direct sums of Rickart modules to be Rickart. In particular, we investigate when free and projective $R$ -modules have the Rickart property and characterize the class of rings for which this happens.

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

THE OHIO STATE UNIVERSITY, LIMA
LIMA, OH
cosmin at math dot ohio-state dot edu

Modules which are isomorphic to their factor modules

ADAM SALMINEN, University of Evansville

Let $R$ be a commutative ring with identity and let $M$ be an infinite unitary module over $R$ . We call $M$ homomorphically congruent (HC for short) provided $M/N\cong M$ for every submodule $N$ of $M$ for which $\vert M/N\vert=\vert M\vert$ . We classify HC modules over Dedekind domains, extending Scott's classification over $\mathbb{Z}$ , and give other results over arbitrary commutative rings.

UNIVERSITY OF EVANSVILLE
EVANSVILLE, IN
as341 at evansville dot edu

The problem of a linear operator with two invariant subspaces

MARKUS SCHMIDMEIER, Florida Atlantic University

For $k$ a field we consider systems $(V, T, U_1, U_2)$ consisting of a finite dimensional $k$ -vector space $V$ , a nilpotent $k$ -linear operator $T:V\to V$ and two subspaces $U_1$ , $U_2$ of $V$ such that $U_1\subset U_2$ .

Each system is a direct sum of indecomposable ones; the complexity of the classification problem for the indecomposable systems increases with the nilpotency index $n$ of $T$ . For $n<4$ , there are only finitely many indecomposable systems, up to isomorphy, while for $n>4$ the classification problem is considered infeasible.

In this talk we discuss vector spaces with two invariant subspaces in the critical case where $n=4$ . In particular, we present a full list of the dimension types of the indecomposable systems. This is a report on joint work with Audrey Moore (Delaware State University).

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

Stone representation theorems for Boolean, potent,

and periodic rings

HANS SCHOUTENS, City University of New York

An element in a ring is called potent (respectively, $n$ -potent) if it equals one of its proper powers (respectively its $n$ -th power), and periodic if two of its powers are equal. A (commutative) ring in which each element is $n$ -potent, potent, or periodic is called respectively, $n$ -Boolean, potent, or periodic. The classical example are Boolean rings (=$2$ -Boolean in our terminology), and Stone proved that a ring is Boolean if and only if it embeds in a power set ring. I will give similar characterizations for the more general rings discussed here. As a corollary, I obtain that a ring is periodic if and only if every finitely generated subring is finite.

CITY UNIVERSITY OF NEW YORK
NEW YORK CITY, NY
hschoutens at citytech dot cuny dot edu

Faithful Torsion Modules

RYAN CRAIG SCHWIEBERT, Ohio University

Call a nonzero R-module "torsion" if all of its elements have nonzero annihilator in R, and "faithful" when the module has zero annihilator in R. Rings admitting torsion modules are completely classified as non-division rings. What rings admit faithful torsion modules? What is the minimum size of a generating set for a faithful torsion module of such a ring? These questions are addressed for several classes of rings, including quasi-Frobenius, unit regular, duo serial, trivial socle, and commutative semiprimitive rings. (Joint work with Greg Oman)

OHIO UNIVERSITY
ATHENS, OH
schwiebert at math dot ohiou dot edu

Small injectivity

LIANG SHEN, Southeast University

A right ideal I of a ring R is called small if, for any right ideal K of R, I+K=R informs K=R. A right R-module M is called small injective if every homomorphism from a small right ideal of R to M can be extended to one from R to M. Some results on small injectivity are given and some known results are improved.

SOUTHEAST UNIVERSITY
NANJING, CHINA
lshen at seu dot edu dot cn

$\Sigma$ -injective Modules: New Characterizations and Applications

ASHISH K. SRIVASTAVA, St. Louis University

A module $M$ is called $\Sigma$ -injective if every direct sum of copies of $M$ is injective. We will present some new characterizations of $\Sigma$ -injective modules. We will also discuss some applications of the study of $\Sigma$ -injective modules.

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