ABSTRACTS

Perturbation of matrices over a quaternion division algebra

SK SAFIQUE AHMAD, Indian Institute of Technology Indore

Matrices over a quaternion division ring and their properties to derive canonical forms of matrices over a quaternion division ring will be derived. Later, some applications for the stability analysis of various systems will be derived with the help of eigenvalues and zeros of polynomials over a quaternion division ring.

INDIAN INSTITUTE OF TECHNOLOGY INDORE
INDORE, MADHYA PRADESH
safique at iiti.ac.in

Matrix wreath products of algebras and embedding theorems

HAMED ALSULAMI, King Abdulaziz University

We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. We also construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimensions over a countable field which answers a question of Bell-Small-Smoktunowicz.

KING ABDULAZIZ UNIVERSITY
JEDDAH, SAUDI ARABIA
hhaalsalmi at gmail.com

Modules satisfying Gabriel's condition H

JOHN A. BEACHY, Northern Illinois University

(Preliminary report, with co-author Yaser Tolooei)

We study modules $M$ that satisfy the analog of Gabriel's condition H in the category $\sigma[M]$.

NORTHERN ILLINOIS UNIVERSITY
DEKALB, IL
johnabeachy at gmail.com

Atoms in Quasi-Local Atomic Domains

KEVIN WILSON BOMBARDIER, University of Iowa

We consider quasi-local integral domains $(R,M)$ in which every nonzero nonunit is a finite product of irreducible elements, also called atoms, and usually with a finite number of non-associate atoms. We discuss several results concerning the number of non-associate atoms, the number of non-associate atoms not in $M^2$, and the number of generators of $M$.

UNIVERSITY OF IOWA
IOWA CITY, IOWA
kevin-bombardier at uiowa.edu

On maps characterized by action on equal products

LOUISA CATALANO, Kent State University

Let $D$ be a division ring with characteristic different from 2. We will describe an additive map $f$ satisfying the identity $f(x)y+xf(y)=l$ for every $x,y\in D$ such that $xy=a$, where $l,a\in D$ and $a$ is nonzero. Additionally, for nonzero $m,k\in D$, we will describe the additive bijection $f$ satisfying the identity $f(x)f(y)=m$ for every $x,y\in D$ such that $xy=k$; this description is the solution to a problem posed by Chebotar, Ke, Lee, and Shiao in 2005.

KENT STATE UNIVERSITY
KENT, OHIO
lcatala1 at kent.edu

Gray map images of constacyclic codes over $\mathbb{Z}_8$

HENRY CHIMAL, Ohio University

$\gamma$-constacyclic codes over $Z_8$ have been characterized in terms of their images under the Gray isometry $\Phi:(\mathbb{Z}_8,\delta_{hom})\rightarrow (\mathbb{F}^{4}_2)$ only when the unit $\gamma$ is either $1$ or $5$. The purpose of this work is to present a characterization of $\gamma$-constacyclic codes in terms of their Gray images when $\gamma=3$ or $5$. To this end, we introduce an isometry $\varphi$ from $(\mathbb{Z}_8,\delta_{hom})$ into $(\mathbb{Z}_4,\delta_L)$ which is a factor of the Gray isometry $\Phi$. Then a connection between $3$- and $7$-constacyclic codes with multi-twisted codes over $Z_4$ is also presented.

OHIO UNIVERSITY
ATHENS, OH
hc118813 at ohio.edu

Structure and distance distributions of repeated-root

constacyclic codes of prime power length over finite fields

HAI Q. DINH, Kent State University

Let p be a prime, and $\lambda$ be a nonzero element of the finite field $\mathbb{F}_{p^m}$. The $\lambda$-constacyclic codes of length $p^s$ over $\mathbb{F}_{p^m}$ are linearly ordered under set-theoretic inclusion, i.e., they are the ideals $\langle(x-\lambda_0)^i\rangle$, $0\leq i \leq p^s$, of the chain ring $\frac{\mathbb{F}_{p^m}[x]}{\langle x^{p^s}-\lambda \rangle}$. This structure is used to establish the Hamming and symbol-pair distances of all such $\lambda$-constacyclic codes. Among others, all MDS symbol-pair constacyclic codes of length $p^s$ are obtained. As an application, we establish all quantum MDS symbol-pair codes of length $p^s$ over $\mathbb{F}_{p^m}$ constructed by CSS construction. These quantum MDS symbol-pair codes are new in the sense that their parameters are different from all the known ones.

KENT STATE UNIVERSITY
WARREN, OHIO
hdinh at kent.edu

Invertible and I2 Algebras

JEREMY ROBERT EDISON, University of Iowa

An algebra $A$ over a field $\mathbb{K}$ is said to be invertible if it has a basis $\mathcal{B}$ consisting exclusively of units. If $A$ has a basis of units $\mathcal{B}$ such that $\mathcal{B}^{-1}$ is again a basis, then $A$ is said to be an invertible-2, or I2, algebra. López-Permouth, Moore, Pilewski, and Szabo showed in 2015 that any finite dimensional invertible algebra $A$ is in fact I2. It is unknown whether an arbitrary invertible algebra is I2. We present here results in this direction for the infinite dimensional case. In particular, we show that if $\mathbb{K}$ is an infinite field and $\mathbb{L}/\mathbb{K}$ is any extension of fields, then $\mathbb{L}$ is an I2 $\mathbb{K}$ algebra. We also present some results suggesting that invertible, finitely generated, commutative $\mathbb{K}$ algebras are, in a sense, “almost I2.”

UNIVERSITY OF IOWA
IOWA CITY, IOWA
jeremy-edison at uiowa.edu

Error Correcting Codes in a Frobenius Algebra Ambient

(Preliminary Report)

ERIK HIETA-AHO, Ohio University

Cyclic codes are among the most studied error-correcting codes. Negacyclic, constacyclic and polycyclic codes are systematic generalizations of cyclic codes. Their underlying common feature is that they can be considered as ideals of certain rings (their Ambient ring.) Cyclic and negacyclic codes share the appealing property that the dual of a cyclic (negacyclic) code is also cyclic (negacyclic) code; in fact the duals are ideals of the same ambient ring. On the other hand, while Constacyclic codes still satisfy that their duals are of the same type, a constacyclic code and its dual are not necessarily ideals of the same ambient ring. The relationship between such pairs of ambient rings has recently been explored in [2]. Noting the fact that the duals of polycyclic codes are not polycyclic [3] and observing the alternative of using annihilators in lieu of dual codes proposed and studied in [1] suggests an alternative approach. We extend the results in [1] by assuming only that the ambient ring is a Frobenius algebra. While Frobenius rings in general satisfy the double annihilator condition and that makes it so that an ideal is completely determined by its annihilator, we have only been successful so far in the context of a Frobenius algebra where the additional structure has allowed us to construct an appropriate balanced non-degenerate bilinear form. We have also managed to obtain analogues to the MacWilliams identities in this setting. [1] Alahmadi, Dougherty, Leroy, and Solé, On the Duality and the direction of polycyclic codes, Advances in Mathematics of Communications 10, (2016), 923-931. [2] Gómez-Torrecillas, Lobillo, and Navarro. Dual Skew Codes from Annihilators: Transpose Hamming ring extensions, preprint, 2017. [3] López-Permouth, Parra-Avila, and Szabo, Dual generalizations of the concept of cyclicity of codes, Advances in Mathematics of Communications 3, (2009), 227-234.

OHIO UNIVERSITY
OHIO, OHIO
eohietaaho at gmail.com

On Leavitt Path Algebras over commutative rings

PRAMOD KANWAR, Ohio University - Zanesville

An ideal (left, right, two sided) $I$ of Leavitt Path Algebra $L_{R}(E)$ over a commutative unital ring $R$ is called basic if for $0\neq r\in R$ and $v\in E^{0}$, $rv\in I$ implies $v\in I$. Among other things, we show that for a finite acyclic graph, the Leavitt Path Algebra is a direct sum of minimal basic ideals and if $R$ has no nonzero nilpotent elements then every minimal basic left ideal $L_{R}(E)x$ contains a vertex. (This is a joint work with Meenu Khatkar and R.K.Sharma)

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

Groups whose nonabelian subgroups satisfy multiple conditions

ZEKERIYA YALÇiN KARATAS,

University of Cincinnati Blue Ash College

The study of groups whose subgroups satisfy certain conditions has a rich and long history. The number of conditions that the subgroups satisfy usually are one or two. However, in some recent papers, the authors determined the structure of groups with three conditions on their subgroups. The first example was given in 2014, which is the structure of groups whose nonabelian subgroups are of finite rank or normal. Some more studies are done since then. In this talk, I will give some results about infinite rank $\mathfrak{X}$-groups in which every nonabelian subgroup is permutable or of finite rank. I will give the definitions, well-known results and some history about these type of problems as well.

*Joint work with Martyn. R. Dixon from the University of Alabama

UNIVERSITY OF CINCINNATI BLUE ASH COLLEGE
BLUE ASH, OH
karatazy at ucmail.uc.edu

$\Sigma$-Rickart modules

GANGYONG LEE, Chungnam National University

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50's. Hereditary rings have been characterized in different ways, the most common of them is that given in [Ch.I, Theorem 5.4, 1]: a ring $R$ is right hereditary if and only if every submodule of any projective right $R$-module is projective if and only if every factor module of any injective right $R$-module is injective.

In this talk, we introduce the notion of $\Sigma$-Rickart modules by utilizing the endomorphism ring of a module and by using the recent notion of Rickart modules [2] as a module theoretic analogue of a right hereditary ring. Note that $M=R_R$ is a $\Sigma$-Rickart module if and only if $R$ is a right hereditary ring. A module $M$ is called $\Sigma$-Rickart if every direct sum of copies of $M$ is Rickart [Definition 2.21, 3]. It is shown that any direct summand and any direct sum of copies of a $\Sigma$-Rickart module are $\Sigma$-Rickart modules. Also, we provide several characterizations of $\Sigma$-Rickart modules which are including generalizations of the most common results (see the above results for rings) of hereditary rings in a module theoretic setting: that is, $M$ is a $\Sigma$-Rickart module if and only if every $M$-generated submodule of any element in Add$(M)$ satisfies $D_2$ condition, if and only if $\mathfrak{E}_M$ is closed under $M$-pure epimorphisms, provided $M$ is $M^{(\mathcal{I})}$-projective for every index set $\mathcal{I}$. Also, we have a characterization of a finitely generated $\Sigma$-Rickart module in terms of its endomorphism ring.

This talk is based on a joint work with Mauricio-Bárcenas.

Bibliography.

[1] H. Cartan; S. Eilenberg, Homological Algebra, Princeton University Press (1956)

[2] G. Lee; S.T. Rizvi; C.S. Roman, Rickart modules, Comm. Algebra, 2010 38(11), 4005-4027

[3] G. Lee; S.T. Rizvi; C.S. Roman, Direct sums of Rickart modules, J. Algebra, 2012 353, 62-78

CHUNGNAM NATIONAL UNIVERSITY
DAEJEON, SOUTH KOREA
lgy999 at cnu.ac.kr

SPLITTING LEGHTH OF ABELIAN MT-GROUPS

EKATERINA KOMPANTSEVA,

Moscow Pedagogical State University,

Financial University under the Government of the RF

The splitting length $l(G)$ of an abelian group $G$ is the smallest natural number $n$ such that the group $\bigotimes\limits^nG=G\otimes\cdots\otimes G$ ($n$ copies) splits. If the group $\bigotimes\limits^nG$ does not split for any natural $n$, then $l(G)=\infty$. The concept of the splitting length of an abelian group was introduced in [1].

In this paper, we consider questions of the splitting length of $MT$-groups. An abelian group $G$ is called an $MT$-group if every multiplication on the torsion part $T(G)$ of the group $G$ uniquely extends to multiplication on $G$. Multiplication on an abelian group $G$ is a homomorphism $\mu:G\otimes G\rightarrow G$. The problem of describing $MT$-groups was formulated in [2, p. 34, Problem 38].

All the groups considered in this work are abelian, and the word “group” will mean “abelian group”. We use the following notation: $\mathbb{N},\mathbb{N}_0$ are sets of possitive and non-negative integers respectively; if $G$ is a group, then $T_p(G)$ is the $p$-component of $G$, $\Lambda (G)$ is the set of prime numbers $p$ such that $T_p (G)\neq 0$; $h_p (g)$ is the $p$-height of the element $g \in G$.

Let $G$ be a group, $d$ be a real number, and $p$ be a prime number. According to [3], the element $g \in G$ satisfies the condition $(\ast)$ for $d$ and $p$ if there exists a non-decreasing unbounded function $f:\mathbb{N}_0\rightarrow\mathbb{N}_0$ such that $h_p(p^ig)>d(i+f(i))$ for every $i \in\mathbb{N}_0$. It is shown in [3] that $l(G)\leq n\;(n\geq 2)$ if and only if for every $g \in G$ there exists $k\in\mathbb{N}$ such that $kg$ satisfies condition $(\ast)$ for $\frac{n}{n-1}$ and every $p\in\Lambda(G)$.

In [4], sets $G^{(n)}\;(n\in\mathbb{N})$ and $G^\ast$ are defined for the group $G$ as follows: $G^{(n)}=\{g\in G\;\vert\;(\exists k\in\mathbb{N})\;kg\;$satisfies the condition$\;(\ast)\;$for$\frac{n}{n-1}\;$and every$\;p\in\Lambda(G)\}$ $(n\geq 2)$, $G^{(1)}=T(G)$, $G^\ast=\bigcup\limits_{n\in \mathbb{N}}G^{(n)}$.

It is shown that $G^\ast$ and $G^{(n)}$ are pure fully invariant subgroups of the group $G$ for any $n\in\mathbb{N}$. Obviously, $T(G)\subseteq G^{(1)}\subseteq G^{(2)}\subseteq\cdots$. In addition, $l(G)\leq n$ if and only if $G=G^{(n)}\;(n\geq 2)$.

It is proved in [5] that if $G$ is an $MT$-group and the quotient group $G/T(G)$ is at most countable, then $l(G)\leq 3$. In the present paper, this result is generalized as follows.

Theorem 1. If $G$ is a $MT$-group and $G/G^{(n)}$ is at most countable $(n\in\mathbb{N})$, then $G=G^{(2n+1)}$.

Theorem 2. If $G$ is a $MT$-group, then either $G/G^{\ast}$ is uncountable or $G=G^\ast$.

References.

[1] Irwin J.M., Khabbaz S.A., Rayna G. Role of tensor product in splitting of abelian groups // J. Algebra, 14 (1970). P. 423-442.

[2] Topics in abelian groups, Chicago, Ill., 1963.

[3] Toubassi E.H., Lawver D.A. Height-slope and splitting length of abelian groups // Publs. Math., 20 (1973). P. 63-71.

[4] Kompantseva E.I. Absolute nil-ideals of an abelian group // J. Math. Sci., 197 (2014). P. 625-634.

[5] Moskalenko A.I. On the splitting lenghth of abelian groups // Mat. Zametki, T. 24, No 6 (1978). P. 749-762.

MOSCOW PEDAGOGICAL STATE UNIVERSITY, FINANCIAL UNIVERSITY UNDER THE GOVERNMENT OF THE RF
MOSCOW, RUSSIAN FEDERATION
Kompantseva at yandex.ru

Rings on quotient divisible abelian groups of rank 1

THI QUYNH TRANG NGUYEN, Moscow Pedagogical State University

Multiplication on an abelian group $G$ is a homomorphism $\mu:G\otimes G\rightarrow G$. The abelian group $G$ with a multiplication on it is called the ring on the group $G$. The set $MultG$ of all multiplications on $G$ paired with the addition is an abelian group. The relation between the structure of an abelian group and the properties of rings on it was studied by L. Fuchs, R. Beaumont, R. Pierce, S. Feigelstock, E. Kompantseva, R. Andruskiewicz and others. An abelian group $G$ is called a $SACR$-group if every ring on $G$ is commutative and associative. If every associative ring with additive group $G$ is filial, then $G$ is called $TI$-group. A ring $R$ is filial if, for any subrings $I, J$ of $R$, $I\lhd J\lhd R$ implies $I\lhd R$. Problems of studying $SACR$-groups and $TI$-groups are formulated in [1, 2]. In these papers, authors described torsion $TI$-groups, torsion part of mixed $TI$-groups and gave some nontrivial examples of torsion-free and mixed $SACR$-groups.

The aim of present work is to study rings on quotient divisible abelian groups. An abelian group $A$ is called a quotient divisible if it does not contain non-zero divisible torsion subgroups, but contains a free subgroup $F$ of finite rank, such that $A/F$ is a divisible torsion group. Torsion-free quotient divisible groups were introduced by R. Beaumont and R. Pierce in [3]. A. Fomin and W. Wickless defined mixed quotient divisible groups and proved that categories of mixed quotient divisible groups and finite-rank torsion-free groups with quasihomomorphisms as morphisms are dual [4]. In [1, 2], it was proved that every torsion-free abelian group of rank 1 is an $SACR$-group and $TI$-group. The duality preserves the torsion-free rank, hence the study of rings on quotient divisible abelian groups should be based on studying rings on these groups of rank 1.

Theorem 1. If $A$ is a quotient abelian group of rank 1, then $Mult A\cong A$.

Theorem 2. Every quotient divisible abelian group of rank 1 is both a $SACR$-group and a $TI$-group.

Bibliography.

[1] Andruszkiewicz R., Woronowicz M. On additive groups of associative and commutative rings // J. Quaest. Math. 2017. V. 40. No 4. P. 527-537.

[2] Andruszkiewicz R., Woronowicz M. On $TI$-groups // Recent Results in Pure and Applied Math. Podlasie, 2014. P. 33-41.

[3] Beaumont R., Pierce R. Torsion free rings // Illinois J. Math., 5 (1961). P. 61-98.

[4] Fomin A., Wickless W. Quotient divisible abelian groups // Proc. Amer. Math. Soc. 1998. V 126, No. 1, P. 45-52.

MOSCOW PEDAGOGICAL STATE UNIVERSITY
MOSCOW, RUSSIAN FEDERATION
trangnguyen.ru at gmail.com

Unique Product Groups and Kaplansky's Conjecture

PACE P. NIELSEN, BYU

We present a new example of non-unique product sets in a torsion-free group. We further explain how the search for such examples has shown that any counterexample to Kaplansky's zero divisor conjecture must have at least 8 elements in the support.

BYU
PROVO, UT
pace at math.byu.edu

The Big Lattice of Lattice Preradicals

SEBASTIAN PARDO GUERRA, UNAM

A lattice preradical is an endofuntor on the category $\mathcal{L_{M}}$ of linear modular lattices, whose objects are the complete bounded modular lattices and whose morphisms are linear morphisms. We have studied the big lattice of lattice preradicals, as well as the four classical operations that occurs in the lattice of preradicals of modules for a ring $R$, namely: the join, the meet, the product and the coproduct. As the lattice of submodules of a module is a complete bounded modular lattice, we show some results of the lattice of module preradicals that extends to the lattice of lattice preradicals, such as the existence of the equalizer, the annihilator, the coequalizer and the totalizer for a lattice preradical $\sigma$.

UNAM
MEXICO CITY, MEXICO CITY
chapospg at gmail.com

Distributive Hierarchies and Graph Induced Magmas

ISAAC OWUSU-MENSAH, Ohio University

A triple $( S, \circ, \ast)$ where $( S, \circ)$ and $( S, \ast)$ are magmas is said to be a distributive magma if $\ast$ left ( resp. right or two-sided) distributes over $\circ$. Given a magma $( S, \circ)$ the thrust of this work is to study the properties inherent in $( S, \ast)$ such that $( S, \circ, \ast)$ is a distributive magma( left, right or two-sided). In [1], left, right, and two-sided distributivity hierarchy graphs of a set are introduced.

Given a set S, its (left, right, two-sided) hierarchy graph (lH(S), rH(S), H(S)) has $M(S)$ as vertices and there is an edge from one operation $*$ to another one $\circ$ if $*$ distributes over $\circ$, respectively, on the left, on the right, or on both sides.

Even if one focuses on finite sets, the complexity of these hierarchy graphs grows very rapidly. Given $* \in M(S)$, the set $\text {out} (*) = \circ \in M(S) \vert * \text {distributes over }\circ \}$ is called the outset of $*$. The graph-theoretic terminology sets the state to formulate several intriguing questions. For example, one can wonder about the possible sizes for the outsets of the various hierarchy graphs on a set $S$. While the combinatorial issues related to the above question seem challenging, the relation between operations in M(S) having similar outsets is another interesting problem, especially if one uses a notion of similarity that is itself of an algebraic nature. The association of these concepts to graph magmas ( operations induced by graphs) are also explored. ( This is a preliminary report on a current project with S. López-Permouth and A. Rafieipour )

[1]. López-Permouth and L. H. Rowen, Distributive hierarchies of binary operations, to appear in Proceedings volume dedicated to the memory of Bruno J. Mueller. Contemporary Mathematics series of the American Mathematical Society

OHIO UNIVERSITY
ATHENS, OH
io103314 at ohio.edu

Algebraic structures on the set of all magmas over a fixed set

ASIYEH RAFIEIPOUR, University of Kashan

In recent years, the word magma has been used to designate a pair of the form $( S, \ast)$ where $\ast$ is a binary operation on the set $S$. Inspired by that terminology, we use the notation $M(S)$ (the magma of $S$) to denote the set of all binary operations on the set $S$ (i.e. all magmas with underlying set $S$.). Given $M(S)$, the set $out(\ast) = \{ \circ \in M(S) \vert \ast distributes over \circ \}$ is called the outset of $\ast$. We define an operation that makes $M(S)$ a monoid in such a way that each outset is a submonoid. This endowment gives us a possibility to compare the various elements of $M(S)$ with respect to the monoid structure of their outsets. Various properties of the operation mentioned above are considered, including multiple additive structures on $M(S)$ that have it as the multiplicative part of a nearring. This is a preliminary report on an ongoing joint project with S.R. López-Permouth and Isaac Owusu Mensah.

UNIVERSITY OF KASHAN
ATHENS, OH
ar996517 at ohio.edu

Amenable bases

ABDULKADER REBIN, Ohio University

A basis $B$ over an infinite dimensional $F$-algebra $A$ is amenable if for all $r\in A$, the set of the coordinate vectors of the family $\{rb\vert b\in B\}$ with respect to $B$ is summable. A basis $B$ is said to be congenial to a basis $C$ if the coordinate vectors of the elements of $B$ represented with respect to $C$ is summable. If $B$ is congenial to $C$ but $C$ is not congenial to $B$, then we say that $B$ is properly congenial to $C$. An amenable basis $B$ is called simple if it is not properly congenial to any other amenable basis. In literature, the fundamental question whether all algebras have simple bases has been raised. In this work, using a construction inspired by that in literature, we introduce a family of algebras granting us examples of algebras without simple bases and of one-sided simple bases. The same construction also provides examples which shows that the notion of amenability is not left-right symmetric. This is a joint work with Pinar Aydogdu and Sergio R. López-Permouth.

OHIO UNIVERSITY
ATHENS, OHIO
rm775311 at ohio.edu

The Schroeder-Bernstein property for modules

S. TARIQ RIZVI, The Ohio State University, Lima

The well known Schröder-Bernstein Theorem states that any two sets with one to one maps into each other are isomorphic. The question of whether any two (subisomorphic or) direct summand subisomorphic algebraic structures are isomorphic, has long been of interest. In this talk, we extend the study of this question for modules. We say that a module $M$ satisfies the Schröder-Bernstein property (S-B property) if any two direct summands of $M$ which are subisomorphic to direct summands of each other, are isomorphic. It is shown that a number of classes of modules satisfy the S-B property. These include the classes of quasi-continuous, directly finite, quasi-discrete and modules with ACC on direct summands. We show that over a Noetherian ring $R$, every extending module satisfies the S-B property. Among applications, it is proved that the class of rings $R$ for which every $R$-module satisfies the S-B property is precisely that of pure-semisimple rings. We show that over a commutative domain $R$, any two quasi-continuous subisomorphic $R$-modules are isomorphic if and only if $R$ is a PID. Examples illustrating the results will be exhibited. (This is a joint work with Najmeh Dehghani and Fatma A Ebrahim)

THE OHIO STATE UNIVERSITY, LIMA
LIMA, OHIO
rizvi.1 at osu.edu

Quasi-Baer Module Hulls and Applications

COSMIN S ROMAN, The Ohio State University

For a module $N$, the quasi-Baer hull qB$(N)$ (resp., the Rickart hull R$(N)$) of $N$ is the smallest quasi-Baer (resp., Rickart) extension of $N$ if it exists, in a fixed injective hull $E(N)$. We initiate the study of quasi-Baer and Rickart module hulls in this research. When a ring $R$ is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective $R$-module has a quasi-Baer hull.

Let $R$ be a Dedekind domain with $F$ the field of fractions. Assume that $M_R$ is an $R$-module with Ann$_R(M)\neq 0$ and $\{K_i\mid i\in\Lambda\}$ is a set of $R$-submodules of $F_R$. Then we show that $M_R\oplus (\oplus_{i\in\Lambda}K_i)_R$ has a quasi-Baer module hull if and only if $M_R$ is semisimple. Also the quasi-Baer hull of $M_R\oplus (\oplus_{i\in\Lambda}K_i)_R$ is explicitly described. An example such that $M_R\oplus (\oplus_{i\in\Lambda}K_i)_R$ has no Rickart module hull is constructed. As a consequence, for a module $N$ over a Dedekind domain with $N/t(N)$ is projective and Ann$_R(t(N))\neq 0$, where $t(N)$ is the torsion submodule of $N$, we show that the quasi-Baer hull qB$(N)$ of $N$ exists if and only if $t(N)$ is semisimple.

We also prove the existence of the Rickart hull R$(N)$ of such an $N$. Furthermore, we provide explicit constructions of qB$(N)$ and R$(N)$ and show that these two hulls are precisely the same. As applications, it is shown that if $N$ is a finitely generated module over a Dedekind domain, then $N$ is quasi-Baer if and only if $N$ is Baer if and only if $N$ is semisimple or torsion-free. Moreover, if $N$ is a module over a Dedekind domain, which is a direct sum of finitely generated modules, it is shown that $N$ is quasi-Baer if and only if $N$ is Rickart if and only if $N$ is semisimple or torsion-free. Examples showing the disparity between Rickart hulls and Baer hulls and illustrating our results will be shown subject to time. (This is a joint work with Gangyong Lee, Jae Keol Park, and S. Tariq Rizvi.)

THE OHIO STATE UNIVERSITY
LIMA, OHIO
cosmin at math.osu.edu

Modules with the smallest possible weak injectivity domain

MARTHA LIZBETH SHAID SANDOVAL MIRANDA,

OHIO UNIVERSITY CENTER OF RINGS AND ITS APPLICATIONS

Given a module $M \in Mod(R),$ $\mathfrak{In}^{-1}(M):=\{N\in Mod(R)\mid M$    is $N-$injective$\},$ denotes the injective domain of $M.$ Recall that a module $M$ is said to be poor if and only if $\mathfrak{In}^{-1}(M) = SSMod(R).$ In [1], it was proved that for every ring $R,$ the category $Mod(R)$ has poor modules. In particular, $$\bigcap \{ \mathfrak{In}^{-1}(M)\mid M\in Mod(R)\}=SSMod(R).$$

On the other hand, given modules $N, M \in Mod(R),$ $M$ is said to be weakly $N$-injective if whenever $\varphi \in Hom_R(N, E(M)),$ there exists $X\leq E(M)$ satisfying that $X\cong M$ and $\varphi(N)\subseteq X.$ A module $M$ is weakly injective if it is weakly $N-$injective for every $N\in fgmod(R).$ See [2].

Given a module $M,$

$${\mathcal{W}\mathfrak{In}^{-1}(M)}:=\{N \in Mod(R) \mid M$$    is weakly $$N-$$injective $$\},$$

denotes the domain of weak injectivity of M.

In this work, we will present some results related to domains of weak injectivity. In particular, we study $$\bigcap \{ \mathcal{W}\mathfrak{In}^{-1}(M)\mid M\in Mod(R)\}$$ looking for some analogous to the situation of poverty but now for weak injectivity case. This is a joint work with Pinar Aydogdu (Hacettepe University), and Sergio López-Permouth (Ohio University)

References: [1] Adel N. Alahmadi, Mustafa Alkan, and Sergio López-Permouth, Poor modules:the opposite of injectivity, Glasgow Mathematical Journal 52 (2010), no. A, 7-17.

[2] S.K Jain and S.R López-Permouth, A survey of theory of weakly injective, Computational Algebra. Lecture Notes in Pure and Applied Mathematics 151 (1994), 205 - 232.

[3] Rafail Alizade, Engin Büyükasik, Sergio R. López-Permouth, and Liu Yang, Poor modules with no proper poor direct summands, Journal of Algebra (2018), -.

OHIO UNIVERSITY CENTER OF RINGS AND ITS APPLICATIONS
ATHENS, OHIO
marlisha at gmail.com

Amenability Profiles of Basic Modules

BENJAMIN Q STANLEY, Ohio University

Let $A$ be a countably infinite dimensional $K$-algebra, where $K$ is a field and let $\mathcal{B}$ be a basis for $A$. We call $\mathcal{B}$ amenable when it is such that $K^\mathcal{B}$ (the direct product indexed by $\mathcal{B}$ of copies of the field $K$) can be made into an $A$-module in a natural way. The matrices that represent multiplication by elements of $A$, with respect to $\mathcal{B}$ must be row and column finite for this to happen. However, for any basis $\mathcal{C}$ we have that the coefficients with this property form a subalgebra of $A$. We examine which subalgebras can be attained in this way. For a given algebra $A$, this collection is called the amenability profile of $A$.

OHIO UNIVERSITY
ATHENS, OH
bs042712 at ohio.edu

Dimension groups with a group action and their realization

LIA VAŠ, University of the Sciences

If a ring is graded by a group $\Gamma,$ its graded Grothendieck group has a natural action of $\Gamma.$ To study such scenario, we consider pre-ordered abelian groups equipped with an action of $\Gamma$ and define appropriate generalizations of simplicial and dimension groups. The classic definitions correspond to the case when $\Gamma$ is trivial.

If $\Gamma$ is a group such that its integral group ring is noetherian, we prove generalizations of the following classic results: every dimension group is isomorphic to a direct limit of a directed system of simplicial groups, and every simplicial/countable dimension group can be realized as the Grothendieck group of a matricial/ultramatricial algebra over a field. We adapt the Realization Problem for von Neumann regular rings to graded rings and list some other problems.

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

On a class of constacyclic codes of length $4p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$

THANG VO, Department of Mathematical Sciences

Let $p$ be a prime such that $p^m\equiv 3 \pmod 4$. For any unit $\lambda$ of $\mathbb{F}_{p^m}$, we determine the algebraic structures of $\lambda$-constacyclic codes of length $4p^s$ over the finite commutative chain ring $\mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$, $u^2=0$. If the unit $\lambda\in \mathbb{F}_{p^m}$ is a square, each $\lambda$-constacyclic code of length $4p^s$ is expressed as a direct sum of an -$\alpha$-constacyclic code and an $\alpha$-constacyclic code of length $2p^s.$ If the unit $\lambda$ is not a square, then $x^4-\lambda_0$ can be decomposed into a product of two irreducible coprime quadratic polynomials which are $x^2+\gamma x+\frac{\gamma^2}{2}$ and $x^2-\gamma x+\frac{\gamma^2}{2}$, where $\lambda_0^{p^s}=\lambda$ and $\gamma^4=-4\lambda_0$. By showing that the quotient rings $\frac{\mathcal R}{\left\langle(x^2+\gamma x+\frac{\gamma^2}{2})^{p^s}\right\rangle}$ and $\frac{\mathcal R}{\left\langle(x^2-\gamma x+\frac{\gamma^2}{2})^{p^s}\right\rangle}$ are local, non-chain rings, we can compute the number of codewords in each of $\lambda$-constacyclic codes. Moreover, the duals of such codes are also given.

DEPARTMENT OF MATHEMATICAL SCIENCES
KENT, OHIO
mvo at kent.edu

On the Hamming Distance of Repeated-Root Constacyclic Codes

XIAOQIANG WANG, Kent State University

Let $p$ be an odd prime, $s$, $m$ be positive integers, $\gamma, \lambda$ be nonzero elements of the finite field $\mathbb{F}_{p^m}$ such that $\gamma^{p^s}=\lambda$. In this talk, we show that the Hamming distances of all repeated-root $\lambda$-constacyclic codes of length $\eta p^s$ can be determined by that of the simple-root $\gamma$-constacyclic codes of length $\eta$, where $\eta$ is a positive integer. As an application, we compute the Hamming distances of all constacyclic codes of length $4p^s$ over $\mathbb{F}_{p^m}$.

KENT STATE UNIVERSITY
KENT, OHIO
waxiqq at 163.com

Utumi Modules

MOHAMED F. YOUSIF, Ohio State University at Lima

A right $R$-module $M$ is called a Utumi Module ($U$-module) if, whenever $A$ and $B$ are submodules of $M$ with $A\cong B$ and $% A\cap B=0,$ there exist two summands $K$ and $T$ of $M$ such that $% A\subseteq ^{ess}K$, $B\subseteq ^{ess}T$ and $K\oplus T\subseteq ^{\oplus }M$. The class of $U$-modules is a simultaneous and strict generalization of three fundamental classes of modules; namely the quasi-continuous, the square-free and the automorphism-invariant modules. In this talk we show that the class of $U$-modules inherits some of the important features of the aforementioned classes of modules. For example, a $U$-module $M$ is clean if and only if it has the finite exchange property if and only if it has the full exchange property. This is a joint work with Yasser Ibrahim of Cairo University.

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

The lattice structure of R-ad for some Artinian serial rings

SERGIO ZAMORA-ERAZO, UAM-Iztapalapa

An abstract class of $R$-Mod is termed Additive Class if it is closed under taking submodules, homomorphic images, and finite direct sums. The conglomerate of all additive classes is denoted by $R$-ad. If ${\mathcal X}$ is any class of modules, then the class of all modules subgenerated by finite direct sums of elements in ${\mathcal X}$, called $ad({\mathcal X})$, is an additive class. In this talk, we define a generalization of $ad({\mathcal X})$ and show its properties for Artinian serial rings that are isomorphic to a finite direct product of Artinian uniserial rings. We present also a complete description of the lattice structure of $R$-ad for Artinian serial rings that are isomorphic to a finite direct product of Artinian uniserial rings.

UAM-IZTAPALAPA
MEXICO CITY, MEXICO CITY
serazo at xanum.uam.mx