Abstracts of 35 Ohio State - Denison Mathematics Conference presentations

Matrix of Rotation Symmetric Boolean Functions and its Eigenvalues

MANUEL EDGARDO ALBRIZZIO, University of Iowa

Digital signatures are an important feature in any encryption/decryption scheme, as it provides a message with integrity, authenticity, and nonrepudiation. The problem occurs when long messages are being exchanged and signatures that are just as long need to be verified. By using hash functions, a "fingerprint" of the message can be used instead of the message itself for verification, making the process computationally inexpensive. If we consider a single iteration of a general hashing algorithm, we see that there may be functions that allow for more efficient evaluation. This is when Rotation Symmetric Boolean Functions were studied more deeply. This talk focuses on a matrix that comes up when investigating these functions, and answering the question of the exact number of eigenvalues of this matrix.

UNIVERSITY OF IOWA
IOWA CITY, IOWA
malbrizzio at uiowa dot edu

Herstein's problem in rings with involution

SHAKIR ALI, Aligarh Muslim University

Let $R$ be a ring with involution $'*'$. Following [I. N. Herstein, Proc. Amer Math. Soc. 8 (1957), 1104-1110], an additive map $D:R\rightarrow R$ is called a Jordan derivation if $D(x^2)=x)x+xD(x)$ holds for all $x\in R$. An additive map $D:R\rightarrow R$ is called a Jordan $*$-derivation if $D(x^2)=x)x^*+xD(x)$ holds for all $x\in R$. For an element $a\in R$, it is easy to verify that the map $D:x\mapsto xa-ax^\ast$ for all $x\in R$, is a Jordan $*$-derivation. Such $D$ is called an inner Jordan $\ast$-derivation (viz.; [S. Ali and N. A. Dar, Comm. Algebra 49(4) (2021), 1422-1430 ] and [T. K. Lee and Y. Zhou, J. Algebra Appl. 13(4)(2014)] for details and recent results).

In this talk, I will review some recent results on Jordan $*$-derivations and its related mappings in rings with involution. Finally, we conclude our talk with some open problems. (This is a joint work with Nadeem A. Dar).

ALIGARH MUSLIM UNIVERSITY
ALIGARH, UTTAR PRADESH, INDIA
shakir dot ali dot mm at amu dot ac dot in

On generalized Lie-type derivations on algebras

MOHAMMAD ASHRAF, Aligarh Muslim University

ALIGARH MUSLIM UNIVERSITY
ALIGARH, UTTAR PRADESH, INDIA
mashraf80 at hotmail dot com

Holonomic modules and 1-generation in the Jacobian Conjecture

VLADIMIR BAVULA, Department of Pure Mathematics, University of Sheffield

I will talk about my recent results that show that the Jacobian Conjecture, the Conjecture of Dixmier and the Poisson Conjecture are questions about holonomic modules for the Weyl algebra $A_n$, the images of the Jacobian maps, endomorphisms of the Weyl algebra $A_n$ and the Poisson endomorphisms are large in the sense that further strengthening of the results on largeness would imply either to prove the conjectures or produce counter examples. A short direct algebraic proof (without reduction to prime characteristic) of equivalence of the Jacobian and Poisson Conjectures is given (this gives a new short proof of equivalence of the Jacobian, Poisson and Dixmier Conjectures). Some generalizations of Gabber's Inequality are given.

DEPARTMENT OF PURE MATHEMATICS, UNIVERSITY OF SHEFFIELD
SHEFFIELD, UNITED KINGDOM
v dot bavula at sheffield dot ac dot uk

On Hereditary Categories of Quiver Representations

RYAN BIANCONI, The University of Iowa

Let $k$ be a field. We show that for any quiver $Q$, not necessarily finite, that the path algebra $kQ$ is hereditary. We also show that in the case when $Q$ is finite the category of locally finite right $kQ$ modules is hereditary.

THE UNIVERSITY OF IOWA
IOWA CITY, IOWA
ryan-bianconi at uiowa dot edu

Trivial Extensions of $K[x,x^{-1}]$ by $M_\infty(K)$

DANIEL BOSSALLER, Baylor University

While directly infinite algebra are, in general, poorly behaved, some order can be restored by leveraging the fact that many directly infinite algebras have an ideal isomorphic to $M_\infty(K)$, the family of countably infinite matrices with only finitely many nonzero entries. In this talk, I will classify all trivial extensions of the Laurent polynomials, $K[x,x^{-1}]$ by $M_\infty(K)$ and give another proof that the Toeplitz-Jacobson algebra cannot be written as a split extension.

BAYLOR UNIVERSITY
WACO, TEXAS
dpb492 at gmail dot com

On the classification of multiplicity-free products of Schur functions

LUISA CARINI, Department MIFT, University of Messina, Italy.

Schur functions are symmetric polynomials introduced by Schur as characters of irreducible representations of the general linear group of invertible matrices. Schur functions can be generated combinatorially using semistandard Young tableaux and form a basis for the ring of symmetric functions. Basically there are three fundamental products on the ring of symmetric functions, namely the ordinary product, the Kronecker product and the plethysm of Schur functions. All these three products of Schur functions are again symmetric functions and can be expressed in terms of Schur functions basis.
We will present the most recent results on various products of Schur functions which are multiplicity-free in the sense that the coefficients which arise in their expansion as a sum of Schur functions are $0, 1$.

DEPARTMENT MIFT, UNIVERSITY OF MESSINA
MESSINA, ITALY
luisa dot carini at unime dot it

Sequences, Skew exponents, and Consequences

AHMED CHERCHEM, Faculty of Mathematics, USTHB

Using exponents of skew polynomials, we introduce the notion of skew period of skew linear recurring sequence. Some properties and examples are presented.

References.

[1] Bouzidi A. D., Cherchem, A., Leroy, A. (2021).
[2] Exponents of skew polynomials over periodic rings. Comm. Algebra. 49 (4):1639–1655.
[3] Cherchem, A., Leroy, A. (2016). Exponents of skew polynomials. Finite Fields Appl. 37:1–13.

FACULTY OF MATHEMATICS, USTHB
ALGIERS, ALGERIA
ahmedcherchem at gmail dot com

Sums of Commuting Nilpotent and Potent Elements

ALEXANDER DIESL, Wellesley College

Ring theorists have long been interested in looking at ways in which elements can be additively decomposed into elements with special properties. In this talk, we will discuss some recent results concerning such decompositions involving nilpotent elements and potent elements (i.e. elements such that $x^n=x$ for some $n$).

WELLESLEY COLLEGE
WELLESLEY, MASSACHUSETTS
adiesl at wellesley dot edu

Repeated-root constacyclic codes of prime power lengths over finite chain rings: algebraic structures, applications, and generalizations

HAI Q. DINH, Kent State University

For any given prime $p$, we study the algebraic structure of repeated-root $\lambda$-constacyclic codes of prime power length $p^s$ over a finite commutative chain ring $R$ with maximal ideal $\langle \gamma \rangle$. It is shown that, for any unit $\lambda$ of the chain ring $R$, there always exists an element $r\in R$ such that $\lambda-r^{p^s}$ is not invertible, and furthermore, the ambient ring $\frac{R[x]}{\langle x^{p^s}-\lambda \rangle}$ is a local ring with maximal ideal $\langle x-r, \gamma \rangle$. When there is a unit $\lambda_0$ such that $\lambda=\lambda_0^{p^s}$, the nilpotency index of $x-\lambda_0$ in the ambient ring $\frac{R[x]}{\langle x^{p^s}-\lambda \rangle}$ is established. When $\lambda=\lambda_0^{p^s}+\gamma w$, for some unit $w$ of $R$, it is shown that the ambient ring $\frac{R[x]}{\langle x^{p^s}-\lambda \rangle}$ is a chain ring with maximal ideal $\langle x^{p^s}-\lambda_0 \rangle$, which in turn provides structure and sizes of all $\lambda$-constacyclic codes and their duals. Among others, self-dual constacyclic codes are provided. We will also discuss some special cases of the chain ring $R$ that were studied in the literature, as well as some generalizations on the lengths of the codes. As an application, the Hamming distance $d_H$, homogeneous distance $d_{hm}$, Lee distance $d_L$, Euclidean distance $d_E$, and Rosenbloom-Tsfasman distance $d_{R-T}$, of all negacyclic codes of length $2^s$ over $\mathbb{Z}_{2^a}$, are completely determined. Open directions for further generalizations will also be discussed.

KENT STATE UNIVERSITY
WARREN, OHIO
hdinh at kent dot edu

Multiplicative lattices and braces

ALBERTO FACCHINI, University of Padua - Department of Mathematics

The multiplicative lattices we will consider are those defined in [Facchini, Finocchiaro and Janelidze, Abstractly constructed prime spectra, Algebra universalis 83(1) (published in February 2022)]. Multiplicative lattices yield the natural setting in which several basic mathematical questions concerning algebraic structures find their answer. We will consider the particular cases of braces [Facchini, Algebraic structures from the point of view of complete multiplicative lattice, to appear, 2022, http://arxiv.org/abs/2201.03295], with a particular attention to some work in progress with Dominique Bourn and Mara Pompili (Huq=ith in braces!).

UNIVERSITY OF PADUA - DEPARTMENT OF MATHEMATICS
PADOVA, ITALIA
facchini at math dot unipd dot it

Reduced Finitary Incidence Algebras and Their Automorphisms

DANIEL HERDEN, Baylor University

Let $I(P)$ denote the incidence algebra of some locally finite poset $(P,\le)$ over some field $F$ and $\sim$ some equivalence relation on the set of generators of $I(P)$. Then $I(P,\sim )$ is the subset of $I(P)$ of all the elements that are constant on the equivalence classes of $\sim$. If $\sim$ satisfies certain conditions, then $I(P,\sim )$ is a subalgebra of $I(P)$ and is called a reduced incidence algebra. We extend this notion to finitary incidence algebras $FI(P)$ for any poset $(P,\le)$. We investigate reduced finitary incidence algebras $FI(P,\sim )$ and determine their automorphisms in some special cases.

BAYLOR UNIVERSITY
WACO, TEXAS
daniel_herden at baylor dot edu

Fixed Lie product preserving mappings on matrix algebras

HAYDEN JULIUS, Youngstown State University

We consider linear mappings on matrix algebras preserving Lie products (i.e., commutators) equal to fixed elements. This preserver problem is a modification of well-known commutativity preserver problems and Lie homomorphism classification problems in linear algebra and ring theory. Surprisingly, bijective linear maps preserving fixed nonzero Lie products need not also preserve commutativity but a complete description is obtainable in some test cases (and turn out to be fairly close to commutativity preservers). However, there are distinct challenges to answering the problem in general. In this talk, we will motivate this problem, present the known results, and discuss some obstacles to answering the problem in general.

YOUNGSTOWN STATE UNIVERSITY
YOUNGSTOWN, OHIO
hjulius at ysu dot edu

Cryptosystems Based On Abelian Subgroups of Matrices

ZEKERIYA YALÇıN KARATAS, University of Cincinnati Blue Ash College

In this talk, a public-key cryptosystem based on some abelian subgroups of the general linear group will be introduced. As a start, an introduction of mathematical cryptography will be presented. Two random elements of an abelian subgroup of lower triangular matrices in GL$(k, \mathbb{Z}_n)$ will be chosen to define automorphisms for encryption. The encryption and decryption will be given explicitly, and an example will be given in the conclusion. This work is a generalization of the cryptosystem introduced by M. Khan and T. Shah in 2015. Also, it is joint work with Erkam Luy from Erciyes University, Turkey.

UNIVERSITY OF CINCINNATI BLUE ASH COLLEGE
BLUE ASH, OHIO
karatazy at ucmail dot uc dot edu

A Journey through Irreducible Polynomials

SUDESH KAUR KHANDUJA, IISER Mohali

Irreducibility of polynomials has a long history. In 1797, Gauss proved that the only irreducible polynomials with complex coefficients are linear polynomials. However, in view of Eisenstein Irreducibility Criterion proved in 1850, for each number $n\geq 1$, there are infinitely many irreducible polynomials of degree $n$ over rationals. We discuss some generalizations of this criterion as well as of the classical Sch $\ddot{\mbox{o}}$nemann Irreducibility Criterion and present a simple proof of Eisenstein-Dumas irreducibility criterion which has been given in 2020.

IISER MOHALI
SAS NAGAR, PUNJAB
sudeshkaur at yahoo dot com

Rings on quotient divisible abelian groups

EKATERINA KOMPANTSEVA, Financial University

Abstract is provided by the co-author, T. Q. T. Nguyen (click this link)

FINANCIAL UNIVERSITY UNDER THE GOVERNMENT OF THE RF.
MOSCOW, RUSSIA
eikompantseva at fa dot ru, kompantseva at yandex dot ru

On rings whose units and nilpotent elements satisfy some equations

M. TAMER KOŞAN, Gazi University

In this presentation, we will discuss the following equations over an associative ring $R$ with identity:

$$U(R)=J(R)$$

$$U(R)=N(R)$$

$$U(R)=J(R)+N(R)$$

$$U(R)=\Delta(R)$$

which are called $UJ$-rings ([3]), $UU$-rings ([1,2]), $UNJ$-rings ([4]) and $\Delta U$-rings ([5,6,7]), respectively, where $U(R)$ is the group of units, $J(R)$ is its Jacobson radical, $N(R)$ is its group of nilpotents and $\Delta(R)=r\in R\vert\forall u\in U(R): r+u \in U(R) \}$ which is a non-unital subring of $R$.

References.

[1] Calugareanu, G. (2015). UU rings. Carpathian J. Math. 31(2):157–163

[2] Danchev, P. V., Lam, T. Y. (2016). Rings with unipotent units. Publ. Math. Debrecen 88:449–466.

[3] Kosan, M. T., Leroy, A., Matczuk, J. (2018). On UJ-rings. Commun. Algebra 46(5):2297–2303.

[4] Kosan, M. T., Quynh, T. C., Zemlicka, J. (2020). UNJ-rings. J. Algebra Appl. 19(09):2050170.

[5] Kosan, M. T., Quynh, T. C., Tai, D. D. (2021). A generalization of UJ-rings. J. Algebra Appl. (in press)

[6] Kosan, M. T., Zemlicka, J. (2021). Group rings that are UJ-rings. Commun. Algebra 49(6), 2370–2377.

[7] Leroy, A., Matczuk, J. (2019). Remarks on the Jacobson radical. In: Rings, Modules and Codes (Contemporary Mathematics), Vol. 727. Providence, RI: Amer Mathematical Society, pp. 269–276

GAZI UNIVERSITY
ANKARA, TURKEY
tkosan at gmail dot com

The rational hull of modules

GANGYONG LEE, Chungnam National University

As we know that a commutative integral domain can be embedded in a field, called its field of fractions. A construction of a “ring of quotients" from a given ring has been studied by a number of authors: Jacobson, Johnson, Stenström and so on. So, rings of quotients play an important role in the ring theory. Utumi gave another construction of a ring of quotients for any ring with zero left annihilator. While a ring $R$ need not have the classical right ring of quotients in general, it always possesses the maximal right (resp., left) ring of quotients, denoted by $Q(R)$ (resp., $Q^{\ell}(R)$), which is a useful tool to characterize the ring $R$.

In this talk, we provide several new characterizations of the maximal right ring of quotients of a ring using the relatively dense property. In addition, we obtain several properties of the rational hull of modules and rationally complete modules. As a ring is embedded in its maximal right ring of quotients, we prove that the endomorphism ring of a module is embedded into the endomorphism ring of the rational hull of the module. We obtain the equivalent condition for the finite direct sum of the rational hulls of submodules to be the rational hull of the direct sum of those submodules. For a right $H$-module $M$ where $H$ is a right ring of quotients of a ring $R$, we provide several sufficient conditions to be End$_R(M)=ext{End}_H(M)$.

CHUNGNAM NATIONAL UNIVERSITY
DAEJEON, SOUTH KOREA
lgy999 at hanmail dot net

Some distances in noncommutative rings

ANDRÉ LEROY, Université d'Artois

We will define different graphs and distances that can be attached to a noncommutative ring. We will study in particular the case of rings of matrices over division rings.

UNIVERSITÉ D'ARTOIS
LENS, HAUT DE FRANCE
andre dot leroy at univ-artois dot fr

Classification of integral domains between a 2-dimensional regular local ring and its quotient field

ALAN LOPER, The Ohio State University, Newark

In this talk I will discuss the work done with a number of my collaborators on the classification of integral domains between a 2-dimensional regular local ring and its quotient field.

THE OHIO STATE UNIVERSITY, NEWARK
NEWARK, OHIO
lopera at math dot osu dot edu

Pseudo core inverses of a sum of morphisms

WENDE LI, School of Mathematics, Southeast University

Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that both $\varphi: X\rightarrow X$ is a morphism of $\mathscr{C}$ with pseudo core inverse $\varphi^{\scriptsize\textcircled{\tiny D}}$ and $\eta: X\rightarrow X$ is a morphism of $\mathscr{C}$ such that $1+\varphi^{\scriptsize\textcircled{\tiny D}}\eta$ is invertible. Let $\alpha=(1+\varphi^{\scriptsize\textcircled{\tiny D}}\eta)^{-1},$ $\beta=(1+\eta\varphi^{\scriptsize\textcircled{\tiny D}})^{-1},$ $\varepsilon=(1-\varphi\varphi^{\scriptsize\textcircled{\tiny D}})\eta\alpha(1-\varphi^{\scriptsize\textcircled{\tiny D}}\varphi),$ $\gamma=\alpha(1-\varphi^{\scriptsize\textcircled{\tiny D}}\varphi)\eta\varphi^{\scriptsize\textcircled{\tiny D}}\beta,$ $\sigma=\alpha\varphi^{\scriptsize\textcircled{\tiny D}}\varphi\alpha^{-1}(1-\varphi\varphi^{\scriptsize\textcircled{\tiny D}})\beta$, $\delta=\eta^{\ast}(\varphi^{\scriptsize\textcircled{\tiny D}})^{\ast}\eta^{\ast}(1-\varphi\varphi^{\scriptsize\textcircled{\tiny D}})\beta.$ Then we present a sufficient condition such that $f=\varphi+\eta-\varepsilon$ has a pseudo core inverse and give the corresponding expression. As an application, we investigate the pseudo core inverse of a sum with radical element in a unital $\ast$-ring. Let $R$ be a unital $\ast$-ring and $J(R)$ its Jacobson radical. If $a$ is pseudo core invertible with the pseudo core inverse $a^{\scriptsize\textcircled{\tiny D}}$ and $j\in J(R)$, we also give a sufficient condition which ensures that $a+j-\varepsilon$ has a pseudo core inverse. Thus, these results generalize the relevant results of core inverses.

The talk reports a joint work with Jianlong Chen and Mengmeng Zhou.

SCHOOL OF MATHEMATICS, SOUTHEAST UNIVERSITY
NANJING, JIANGSU PROVINCE, CHINA
wendly155 at 163 dot com

Continuity with respect to semisimple submodules

SANJEEV KUMAR MAURYA, University of Allahabad, India

In this paper we introduce and study the class of semisimple continuous modules as a non trivial generalization of continuous module. We discuss some properties and characterization of semisimple continuous module. We compare simple continuous with semisimple continuous module. We characterize the rings over which every module is semisimple continuous. We also characterize the rings over which every semisimple continuous module is quasi-injective.

UNIVERSITY OF ALLAHABAD, PRAYAGRAJ
UTTAR PRADESH, INDIA
sanjeevm50 at gmail dot com

Baer and Rickart lattices

MAURICIO MEDINA-BÁRCENAS, Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México

We introduce the notion of Rickart and Baer lattices. We show that part of the theory of Rickart and Baer modules can be understood just using techniques from the theory of lattices. For, we use linear morphisms, introduced by T. Albu and M. Iosif. We also show that Rickart and Baer lattices can be characterized by the annihilators in the monoid of linear endomorphisms as in the case of modules. Joint work with Hugo Rincón Mejía

DEPARTAMENTO DE MATEMÁTICAS, FACULTAD DE CIENCIAS, UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
MEXICO CITY, MEXICO
mmedina at ciencias dot unam dot mx

Infinite-Dimensional Triangularization

ZAK MESYAN, University of Colorado

One of the most commonly used tools in linear algebra, both theoretically and computationally, is the fact that every matrix with entries from an algebraically closed field is similar to an upper-triangular matrix (with possible additional structure, such as the Jordan canonical form). I will describe a way to generalize the notion of “upper-triangular" to linear transformations of an arbitrary vector space over any field, and generalize some of the usual facts about triangular matrices to that context, in a way to preserves much of the intuition from the finite-dimensional setting. Among other topics, I will discuss the Jordan canonical form, nilpotent transformations, simultaneous triangularization of commuting transformations, and triangularizable algebras.

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

Bounded factorization property for some class of noetherian domains.

ZAHRA NAZEMIAN , University of Graz

Beside several techniques to find examples of noetherian domains with bounded factorization (BF) property, we see that noetherian domains possessing a particular finite partitive function from finitely generated modules to a set of ordinal numbers are BF. This implies that noetherian rings with Auslander dualizing complex are BF. Some examples of these rings are all known noetherian Hopf algebra (including the group ring $kG,$ where $k$ is a field and $G$ is a polycyclic-by-finite group) and Weyl algebras $A_n(k),$ where $k$ is field of characteristic zero. This is some part of a joint work with J. Bell, K. Brown and D. Smertnig.

UNIVERSITY OF GRAZ
GRAZ, AUSTRIA
z_nazemian at yahoo dot com

Extending an algebra structure to arbitrary formal linear combinations of elements of its basis, preliminary report.

AARON TAYLOR NICELY, Ohio University

It was observed before that the ring structure of the ring of Laurent polynomials $F[x,x^{-1}]$ cannot be extended in a natural way to the $F$-vector space consisting of (possibly infinite) linear combinations of positive and negative powers of x. Similar situations were confronted earlier in the literature when the module structure of an algebra A with basis B was to be extended to the F-vector space of arbitrary (possibly infinite) linear combinations of the elements of B. The progress attained in those earlier works was due to the realization that the solvability of the problem depended on a property of the basis B. A basis B that allows the extension is called an amenable basis. Following that lead, we analyze to what extent the extension of the algebra extension may depend on a suitable property of the basis.

This is a preliminary report on a collaboration with Sergio R. Lopez-Permouth.

OHIO UNIVERSITY
ATHENS, OHIO
an636517 at ohio dot edu

Lifting periodic elements

PACE P. NIELSEN, Brigham Young University

An element $x$ is periodic if $x^m =x^n$ for some natural numbers $m>n$. Idempotents are the classical example, where $m=2$ and $n=1$. Lifting idempotents modulo ideals, especially nil ideals, allows one to transfer ring structure from a factor ring back to the original. We discover that more is true, by investigating situations where periodic elements do (or do not) lift modulo ideals, as well as investigating the relationship between periodic element lifting and idempotent lifting.

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

Rings on quotient divisible Abelian groups

TRANG THI QUYNH NGUYEN, FPT University

This is joint work with professor E. I. Kompantseva of Financial University under the Government of the RF.

A multiplication on an Abelian group $G$ is a homomorphism $\mu:G\otimes G\rightarrow G$. A multiplication $\mu:G\otimes G\rightarrow G$ on a group $G$ is often denoted by the symbol $\times$, i.e. $\mu(g_1\otimes g_2)=1\times g_2$ for all $g_1, g_2\in G$. An Abelian group $G$ with a multiplication on it is called a ring on $G$ which is denoted by $(G,\times)$. The set $MultG$ of all multiplications on $G$ itself is an Abelian group with respect to addition; the group is called the multiplication group of $G$.

The problem of studying the relationship between the structure of an Abelian group and the properties of ring structures on it is very multifaceted. R. Andruszkiewicz and M. Woronomicz have formulated the problem of studying filial rings on Abelian groups. An Abelian group $G$ is called a $TI$-group (from "transitive ideal") if every associative ring on $G$ is filial. An associative ring $R$ is filial if for any subrings $I,~J$ of $R$, $I\lhd J\lhd R$ implies $I\lhd R$. Filial rings wereintroduced in [2] and $TI$-groups were studied in [1, 3, 4, 5].

The aim of present work is to study rings on quotient divisible Abelian groups. Torsion-free quotient divisible groups were introduced by R. Beaumont and R. Pierce in [6] when describing groups admitting a ring structure embedded in a semisimple separable algebra. 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 [7]. An Abelian group $G$ 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 $G/F$ is a divisible torsion group. By the basis and rank of a quotient divisible group $G$ we mean any basis and rank of a free such group $F$.

Given an element $g$ of the Abelian group $G$ and a prime number $p$, define $m_p$ as the smallest non-negative integer such that the element $p^{m_p}g$ is divisible by any power $p$ in the group $G$. If there does not exist such number, we put $m_p =nfty$. The sequence $(m_{p_1},m_{p_2},\cdots, m_{p_n},\cdots)$ is called the cocharacteristic of the element $g$ in the group $G$ and is denoted by $cochar(g)$. For a quotient divisible Abelian group $G$ of rank 1, $cochar(G)$ denotes the cocharacteristic of the group $G$ and it is a cocharacteristic of any basic element of $G$. As usual, $\chi(g)$ denotes the characteristic of the element $g$.

In this work the multiplication groups of reduced quotient divisible Abelian groups of rank 1 are described. This allows us possible to study the properties of rings on such groups, in particular, to describe $TI$-groups in this class.

Theorem 1. Let $G$ be a quotient divisible Abelian group of rank 1 with a basis $\{e\}$.

1. For every element $g\in G$ there exists an unique ring $(G,\times)$, in which $e\times e =$
2. Two rings $(G,\times)$ and $(G,\circ)$ are isomorphic if and only if $\chi(e\times e)=hi (e\circ e).$

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

Theorem 2. A reduced quotient divisible Abelian group $G$ of rank 1 is a $TI$-group if and only if $cochar(G)$ does not contain positive integers greater than 1.

References.

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

[2] Ehrlich G. Filial rings // Portugal. Math. 1983-1984. Vol. 42. P. 185–194.

[3] Kompantseva E. I., Nguyen T. Q. T. Algebraically compact abelian $TI$-groups // Chebyshevskii Sb. 2019. Vol. 20. no 1. P. 202–211. (Russian)

[4] Kompantseva E. I., Nguyen T. Q. T., Gazaryan V. A. Filial rings on direct sums and direct products of torsion-free abelian groups // Chebyshevskii Sb. 2021. Vol. 22. no 1. P. 200–212. (Russian)

[5] Kompantseva E. I., Tuganbaev A. A. Rings on Abelian Torsion-Free Groups of Finite Rank // Beitrage zur Algebra und Geometrie (Springer). 2021. P. 1–19.

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

[7] Fomin A. A., Wickless W. Quotient divisible abelian groups // Proc. Amer. Math. Soc. 1998. Vol. 126. no 1. P. 45–52.

FPT UNIVERSITY
HANOI, VIETNAM
trangntq26 at fe dot edu dot vn

Green's Relations, Idempotents, and Regular Elements in the Monoid on the Set of All Binary Operations Over a Fixed Set

ASIYEH RAFIEIPOUR, Ohio University

Let $S$ be an arbitrary set and $\mathcal{M}(S)$ the set of all binary operations on $S$. Our study concerns a monoid structure $(\mathcal{M}(S), \triangleleft )$ where $\triangleleft (\ast , \circ) (a,b) =irc (\ast(a,b), \ast(b, a))$ for all $\ast, \circ \in \mathcal{M}(S)$ and $a, b \in S$. In this presentation, we report on work on the Green's right and left relations over this monoid. In particular, we show that right Green's classes are equal to right associates. Also, we give upper and lower bounds for the size of Idempotent elements and at the end, we will talk about regular elements and show that there always exist some non-regular elements in $\mathcal{M}(S)$ which leads to nonregularity of the monoid $\mathcal{M}(S)$.
(This is part of joint work with S. R. López-Permouth and Isaac Owusu-Mensah for our recent paper in Semigroup Forum Journal.)

OHIO UNIVERSITY
ATHENS, OHIO
ar444818 at ohio dot edu

Rudimentary rings

COSMIN Ş. ROMAN, The Ohio State University, Lima

In 2014, together with my co-authors G. Lee and X. Zhang, we defined the concept of rudimentary ring, which is a ring admitting a faithful module whose endomorphism ring is a division ring. This notion properly generalizes the concept of primitive rings, and is tied in with the problem of when is the converse of Schur Lemma true (and the corresponding condition, dubbed CSL, for module categories). In this presentation I will present results concerning rudimentary rings, some examples as well as a few open questions.

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

Jacobson's lemma and Cline's formula for generalized inverses in a ring with involution

GUIQI SHI, Southeast University

In this talk, we will discuss the Jacobson's lemma and Cline's formula for generalized inverses in a ring with involution. We first recall some corresponding results in the cases of Drazin inverses and generalized Drazin inverses. Then we focus on the necessary and sufficient conditions for $1-ba$ (resp., $ba$) being pseudo core invertible while $1-ab$ (resp., $ab$) has pseudo core inverse. At last, Jacobson's lemma for Moore-Penrose inverses is also considered.

SOUTHEAST UNIVERSITY
NANJING, JIANGSU, CHINA
sgq112358 at 163 dot com

Iterated differential polynomial rings over locally nilpotent rings

JOOYOUNG SHIN, Kent State University

Recently, there has been significant progress in the study of differential polynomial rings. We will survey some of these results. In particular, iterated differential polynomial rings over locally nilpotent rings will be presented. We will discuss that a large class of iterated differential polynomial rings over locally nilpotent rings are Behrens radical. This extends results of Chebotar [1] and Chen, et al [2].

References.

[1] M. Chebotar. On differential polynomial rings over locally nilpotent rings, Israel J. Math. 227 (2018), 233-238.

[2] F. Y. Chen, H. Hagan, and A. Wang. Differential polynomial rings in several variables over locally nilpotent rings, Internat. J. Algebra Comput. 30 (2020), 117-123.

[3] S. Jin, J. Shin. Iterated differential polynomial rings over locally nilpotent rings, Comm. Algebra 49 (2021), 256-262.

KENT STATE UNIVERSITY
KENT, OHIO
jshin5 at kent dot edu

MacWilliams extending conditions and quasi-Frobenius rings

ASHISH K SRIVASTAVA, Saint Louis University

MacWilliams proved that every finite field has the extension property for Hamming weight which was later extended in a seminal work by Wood who characterized finite Frobenius rings as precisely those rings which satisfy the MacWilliams extension property. In this paper, the question of when is a MacWilliams ring quasi-Frobenius is addressed. It is proved that a right or left noetherian left 1-MacWilliams ring is quasi-Frobenius. We also prove that a right perfect, left automorphism-invariant ring is left self-injective. In particular, this yields that if $R$ is a right (or left) artinian, left automorphism-invariant ring, then $R$ is quasi-Frobenius. (Joint work with Pedro A. Guil Asensio)

SAINT LOUIS UNIVERSITY
SAINT LOUIS, MISSOURI
ashish dot srivastava at slu dot edu

Determinant-like maps and $d$-partitions of the complete graph $K_{2d}$

MIHAI DORU STAIC, Bowling Green State University

In this talk we introduce a linear map $det^{S^2}:V_d^{\otimes d(2d-1)}\to k$ that is a natural generalization of the usual determinant. We discuss some properties of $det^{S^2}$, connections to combinatorics, and give a geometrical interpretation for the condition $det^{S^2}((v_{i,j})_{1\leq i<j\leq 2d})=$. These results are joint work with Steven Lippold, Alin Stancu, and Jacob Van Grinsven.

BOWLING GREEN STATE UNIVERSITY
BOWLING GREEN , OHIO
mstaic at bgsu dot edu

On Rings with the Serial Embedding Property

LE TANG, The University of Iowa

A ring is said to be uniserial (resp. serial) if it is a uniserial (resp. serial) module both as a left and right module over itself. A well-know result says that a ring is Artinian serial if and only if all its left modules are serial. It is then natural to ask about the existence of rings such that all (left) modules over it can be embedded in serial modules. We call such property the (left) serial embedding property, or simply SEP. Tuganbaev answered this question in the semiprime case. In this talk, we give a report on our research related to the SEP. This is joint work in process with Miodrag Iovanov and Victor Camillo.

THE UNIVERSITY OF IOWA
IOWA CITY, IOWA
le-tang at uiowa dot edu

On modules in which every finitely generated submodule is a kernel of an endomorphism

YASER TOLOOEI, Razi University

We study an $R$-module $M$ in which every finitely generated submodule of $M$ is a kernel of an endomorphism of $M$. Such modules are called Co-epi-finite-retractable (CEFR). We also consider CEFR condition on the injective hull of simple modules, submodules and factors of a CEFR module, and directsum of CEFR modules. Among other results, we prove that the injective hull of a simple module over a commutative Noetherian ring is uniserial if and only if it is CEFR. We investigate modules over a principal ideal ring and show that all finitely generated torsion modules over a principal ideal domain are CEFR. Also, we show that every module over a commutative Köthe ring is CEFR. We also observe that a ring $R$ is left pseudo morphic if and only if it is CEFR as a left $R$-module and we obtain some new properties of left pseudo morphic rings.

RAZI UNIVERSITY
KERMANSHAH, IRAN
yasertoloei at yahoo dot com

Cancellation properties of graded and nonunital rings

LIA VAŠ, University of the Sciences in Philadelphia

The talk is motivated by two questions “If $P$ is a property of unital rings, how does one define a generalized version of $P$ suitable for nonunital rings?” and “If $P$ is a property of rings, how does one define a graded version of $P$ in a meaningful way?”.

We address these questions for several different properties $P$ related to cancellation. We say that a ring property is a cancellation property if it can be directly related to one of the cancellation properties of modules (internal cancellation, substitution, module-theoretic direct finiteness, and module-theoretic exchange) or if it is “sandwiched” between two properties directly relatable to module cancellation. Thus, being unit-regular, having stable range one, being directly finite, being exchange, and being clean are cancellation properties of rings. We explore the relationships of graded or nonunital (or both) generalizations of these properties.

UNIVERSITY OF THE SCIENCES IN PHILADELPHIA
PHILADELPHIA, PENNSYLVANIA, US
l dot vas at usciences dot edu

The number of codewords in cyclic codes of length $5p^s$ over $\mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ and their duals

MANH THANG VO, Mathematics Department Ohio University

Besides investigating all ideals of $\frac{\mathcal R[x]}{\langle (x^4+x^3+x^2+x+1)^{p^s}\rangle}$ to study detail structure of a cyclic code of length $5p^s$ over $\mathcal R$ and their duals, we give the number of codewords in each of those cyclic codes of length $5p^s$ over $\mathcal R$. As cyclic and negacyclic codes of length $5p^s$ over $\mathcal R$ are in a one-by-one equivalent via the ring isomorphism $x \mapsto -x$, all our results for cyclic codes hold true accordingly to negacyclic codes.

MATHEMATICS DEPARTMENT OHIO UNIVERSITY
ATHENS, OHIO
mv294721 at ohio dot edu

On (b, c)-inverses and (c, b)-inverses of elements in rings

CANG WU, School of Mathematics, Southeast University

The $(b,c)$-inverse is an important generalized inverse since it encompasses a large class of generalized inverses such as the famous Moore-Penrose inverse and Drazin inverse. Many properties and characterizations of $(b,c)$-inverses have been achieved for the case $b=c$, but some of them fail when $b \neq c$. In this talk, we show that the main results of $(b,b)$-inverses can be generalized if an element has both a $(b,c)$-inverse and a $(c,b)$-inverse. As applications, some new characterizations of weighted Moore-Penrose inverses and core-EP inverses of complex matrices are given.

SCHOOL OF MATHEMATICS, SOUTHEAST UNIVERSITY
NANJING, CHINA
wucang95 at 163 dot com, cang_wu at hotmail dot com

On Three Types of Galois Extensions which are Azumaya

LARRY LIANYONG XUE, Bradley University

In this paper, we shall study three types of Galois extensions $B$ which are Azumaya algebras over its center $C$. (1) ${\cal A}_1$, the class of Azumaya Galois extensions; (2) ${\cal A}_2$, the class of Galois extensions $B$ of $B^G$ with Galois group $G$ such that $B^G$ is a separable $C^G$-algebra; and (3) ${\cal A}_3$, the class of Galois extensions which are Azumaya algebras over its center $C$. Clearly, ${\cal A}_1\subseteq {\cal A}_2\subseteq {\cal A}_3$. Examples are given to show the proper inclusion relationship ${\cal A}_1\subset{\cal A}_2\subset {\cal A}_3$, and equivalent conditions are given under which a $B\in{\cal A}_3$ is in ${\cal A}_1$ and a $B\in{\cal A}_2$ is in ${\cal A}_1$.

BRADLEY UNIVERSITY
PEORIA, ILLINOIS
lxue at bradley dot edu

Modules with the Exchange Property

MOHAMED YOUSIF, The Ohio State University at Lima

A module $M$ is said to satisfy the (full) exchange property if for any two direct sum decompositions $L = M\oplus N = \oplus_{i\in \mathcal I}N_i$, there exist submodules $K_i\subseteq N_i$ such that $L=M\oplus (\oplus_{i\in\mathcal I} K_i$. If this holds only for $\vert \mathcal I\vert <\infty$, then $M$ is said to satisfy the finite exchange property. The exchange property is of importance because it provides a way to build isomorphic refinements of different direct sum decompositions, which is precisely what is needed to prove the famous Krull-Schmidt-Remak-Azumaya Theorem. It is an open question due to Crawley and Jónsson whether the finite exchange property always implies the full exchange property. In this talk we present the latest results on this open question and its relationship with clean rings and modules. This is a joint work with Yasser Ibrahim of both Taibah and Cairo Universities.

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

Symmetry Questions About Amenability of Bases of Infinite Dimensional Algebras

MAJED HUSAIN ZAILAEE, Ohio University - King Abdulaziz University

A basis B for an infinite-dimensional algebra A over a field F is said to be left amenable if we can extend the left A-algebra structure on the F-vector space $\bigoplus\limits_{b \in \mathcal{B}}Fb={(\mathcal{B})}$, arising from the correspondence between $F^{\mathcal{B}}$ and $A$, in a natural way to the $F$-vector space $\prod\limits_{b \in \mathcal{B}} Fb={\mathcal{B}}$. The definition of when a basis is right amenable is obtained via the obvious modifications.

Mimicking classical ring-theoretic terminology, we say that an algebra is left duo-amenable (or simply left duo when there is no danger of confusion) if every left amenable basis of A is right amenable. Similarly, an algebra is right-duo if every right amenable basis is left amenable. An algebra that is both left and right amenable is simply said to be duo. Clearly, commutative algebras are duo. As a first step to understand the non-commutative situation, these notions are investigated here, and we focus on graph magma algebras. Graph magma algebras have a solid record as a fertile ground for amenability questions. In this setting we produce examples of graph magma algebras which are right or left duo without being commutative. On the other hand, we prove that a (two-sided) duo graph magma algebra is commutative. We do not know at this point any examples of non-commutative duo algebras.

In addition, in this presentation we characterize left and right amenable bases on graph magma algebras in terms of fixed basis of vertices, describe graph magma algebras that have simple bases, and characterize graph magma algebras that have right simple bases which are not left simple. We see the impact of duo properties on the symmetry question for simplicity.

OHIO UNIVERSITY- KING ABDULAZIZ UNIVERSITY
ATHENS, OHIO
mz576413 at ohio dot edu