Abstracts

Leavitt Path Algebra over Kronecker Square of a Quiver.

JEHAN ABDULLAH ALARFAJ, Saint Louis University

The Kronecker product of quivers was defined by Weichsel. He describes the Kronecker product of two quivers $Q_1$ and $Q_2$ as a quiver $Q_1 \otimes Q_2$ whose adjacency matrix $A_{Q_1\otimes Q_2}$ is the Kronecker product of adjacency matrices $A_{Q_1}$and $A_{Q_2}$. Recall that if $A_{Q_1}=_{ij}]_m$ and $A_{Q_2}=_{ij}]_n$, then the Kronecker product of these matrices is given by $A_{Q_1\otimes Q_2}=_{ij}]_{mn}$ where $c_{ij}={ij} [b_{ij}]$. The Kronecker product of a quiver $Q$ with itself, $Q \otimes Q$, is called the Kronecker square of $Q$ and denoted as $\widehat Q$. The utility of the Kronecker product in neural networks is demonstrating its effectiveness in generating graphs that accurately capture the structural characteristics of real networks. Therefore we study the Leavitt path algebra over Kronecker square of a quiver. We show that some of the graph-theoretic properties are shared by Q and $\hat{Q}$, indeed it can easily be observed that the two Leavitt path algebras $L_K(Q)$ and $L_K(\hat{Q})$ share many ring-theoretic properties. However, we give some examples to show that there are some other ring-theoretic properties with respect to which $L_K(Q)$ and $L_K(\hat{Q})$ show different behaviors. Also, we show that if we have two quivers whose leavitt path algebra are isomorphic, the leavitt path algebra over their Kronecker square would not be isomorphic, however, we prove that the converse of this statement is true.

SAINT LOUIS UNIVERSITY
SAINT LOUIS , MO
Jehan dot alarfaj at slu dot edu

Structure of certain periodic Rings and Near rings

ASMA ALI, Department of Mathematics, Aligarh Muslim University,

Aligarh, India

Using commutativity of rings satisfying (xy)n(x;y) = proved by Searcoid and MacHale [1], Ligh and Luh [2] have given a direct sum decomposition for rings with the mentioned condition. Further Bell and Ligh [3] sharpened the result and obtained a decomposition theorem for rings with the property xy =y)2f(x; y) where f(X; Y ) 2 Z < X; Y >; the ring of polynomials in two noncommut- ing indeterminates. In the present paper we continue the study and investigate structure of certain rings and near rings satisfying the following condition which is more general than the mentioned conditions : xy =x; y); where p(x; y) is an admissible polynomial in Z < X; Y > : Moreover we deduce the commutativity of such rings. References [1] M. O. Searcoid and D. MacHale, Two elementary generalizations of Boolean rings, Amer. Math. Monthly, 93 (1986), 121-122. [2] S. Ligh and J. Luh, Direct sum of J-rings and zero rings, Amer. Math. Monthly, 96 (1989), 40-41. [3] H.E. Bell and S. Ligh, Some decomposition theorems for periodic rings and near rings, Math., J. Okayama Univ., 31 (1989), 93-99.

DEPARTMENT OF MATHEMATICS, ALIGARH MUSLIM UNIVERSITY, ALIGARH, INDIA
ALIGARH, UTTARPRADESH
asma dot ali345 at gmail dot com
asma ali2 at redi mail dot com

On the $n^{th}$-power property and its related maps in rings

SHAKIR ALI, Department of Mathematics, Aligarh Muslim University, Aligarh, India

and Institute of Mathematical Sciences, Universiti Malaya, Malaysia

Let $n\geq 2$ be a fixed integer and $R$ be a ring. An additive mapping $d:R \to R$ is said to be a derivation on $R$ if $d(xy)=d(x)y+xd(y)$ holds for all $x,y\in R$. An additive mapping $d:R \to R$ is said to be a Jordan derivation if $d(x^2)=d(x)x+xd(x)$ holds for all $x\in R$. For $n\geq 2$, it is easy to show (by induction) that if $d$ is a derivation of a ring $R$, then $d$ satisfying the following relation

$$d(x^n)=\sum\limits_{i=0}^{n-1} x^{i}d(x)(x)^{n-i-1}~\mbox{for~ all}~ x\in R$$

where $x^0y = y = yx^0$ for all $x,y \in R.$ This equation is called the “$n^{th}$-power property". The study of such mappings were initiated by Bridges and Bergen [1]. In 1984, they proved that such type of map exhibiting $n^{th}$ power property is a derivation on $R$, when $R$ is a prime ring with identity and when $char~R > n$ or is zero. In the year 2007, Lanski [4] generalized this result from derivations to generalized derivations in semiprime rings. Recently, author together with Dar [2] introduced the notion of “$n^{th}$-power *- property" and studied these results in the setting of rings with involution. Precisely, an analogous result for Jordan *-derivations on prime rings with involution was obtained by Dar and Ali [2] (see also [3] for more related results).

In this talk, we will discuss the recent progress made on the $n^{th}$-power property and related concepts. Moreover, some examples and counter examples will be discussed for questions raised naturally. Finally, we conclude our talk with some open problems.

References

[1] Bridges, D., Bergen, J. (1984). On the derivation of $x^n$ in a ring. Proc. Amer. Math. Soc., 90, 25–29.

[2] Dar, N. A., Ali, S. (2021). On the structure of generalized Jordan $*$-derivations of prime rings, Commun. Algebra, 49(4), 1422–1430.

[3] Jeelani, M., Alhazmi, H., Singh, K. P. (2021). On $n^{th}$ power $*$-property in $*$-rings with applications, Commun. Algebra, 49(9), 3961–3968.

[4] Lanski, C. (2007). Generalized Derivations and $n^{th}$ Power Maps in Rings, Commun. Algebra, 35(11), 3660–3672.

DEPARTMENT OF MATHEMATICS, ALIGARH MUSLIM UNIVERSITY, ALIGARH, INDIA
AND
INSTITUTE OF MATHEMATICAL SCIENCES, UNIVERSITI MALAYA, MALAYSIA
shakir dot ali dot mm at amu dot ac dot in

The Isomorphism Problem For Basic Modules

PINAR AYDOGDU, Hacettepe University

Basic modules, a new kind of module over an infinite dimensional algebra $A$, were introduced in [1]. The name is an intentional pun inspired by the fact that the first step in the construction of these modules is to fix a suitable basis for $A$ over which the entire construction revolves. Not all bases allow the construction to work and, for that reason, the notion of amenability of bases $\mathcal{B}$ of an infinite dimensional algebra $A$ over a field $F$ was also introduced in [1]. Amenability is a condition that permits the regular $A$-module structure on the direct sum $F^{(\mathcal {B})}\cong A$ to be extended, in a natural way, to an $A$-module structure on the direct product $F^{\mathcal {B}}$. That module is hence denoted $F^{\mathcal {B}}=_{\mathcal {B}}\mathcal{M}$ and referred to as the basic module induced by the basis $\mathcal{B}$.

While mutual congeniality of bases is known to guarantee that basic modules from so-related bases are isomorphic, the question of what can be said about isomorphism of basic modules in general has remained open. We show that, for some algebras, basic modules may be non-isomorphic. We also show that it is possible, for some algebras, for all basic modules to be isomorphic, regardless of congeniality.

Also, we introduce the notion of domains of divisibility of modules over arbitrary rings. The mechanism employed here to differentiate non-isomorphic basic modules is by showing that they have different domains of divisibility. Domains of divisibility measure how divisible a module can be; for those cases when divisibility is equivalent to injectivity, domains of divisibility provide a way to gauge injectivity of modules as an alternative to the domains of injectivity and other mechanisms in the literature.

This is a joint work with C.A. Arellano, S.R. Lopez-Permouth, R. Muhammad and M. Zailaee. The speaker was supported by The Scientific and Technological Research Council of Türkiye (TÜBİTAK) with Grant No. 122F105.

[1] L.M. Al-Essa, S. R. López-Permouth, N.M. Muthana, Modules over infinite-dimensional algebras, Linear and Multilinear Algebra 66 (3), 488-496 (2018)

HACETTEPE UNIVERSITY
ANKARA,
paydogdu at hacettepe dot edu dot tr

Universal localization at semiprime Goldie ideals

JOHN A BEACHY, Northern Ilinois University

The localization of a commutative ring at a prime ideal has proven to be a crucial tool in commutative algebra and algebraic geometry. Finding an analog for noncommutative rings has been a goal since the 1960's. Unfortunately, the direct analog often fails to exist, reducing the theory to partial analogs or particular cases. The universal localization, as defined by P.M.Cohn, always exists, but is quite difficult to compute because of its definition via a universal property. Nevertheless, it still seems to have the potential for some important applications

NORTHERN ILINOIS UNIVERSITY
COLORADO SPRINGS, COLORADO
johnabeachy at gmail dot com

On relative homological dimensions

DRISS BENNIS, Mohammed V University in Rabat

In recent years, there has been a growing interest in exploring homological algebra relative to some subcategories. This approach is particularly useful as it offers a unified framework to understand and further develop various well-known homological dimensions. In this talk, we will provide an overview of this trend and propose new directions for future research.

MOHAMMED V UNIVERSITY IN RABAT
RABAT,
driss dot bennis at fsr dot um5 dot ac dot ma

Von Neumann regular and strongly regular matrices

IULIA-ELENA CHIRU, Babes-Bolyai University

Faculty of Mathematics and Computer Science

We give an intrinsic characterization for an $m\times n$-matrix $A$ to be von Neumann regular over a commutative local ring. Specifically, we prove that $A$ is von Neumann regular if and only if $A$ has an invertible $\rho(A)\times \rho(A)$-submatrix, with $\rho(A)$ being the determinantal rank of $A$. An important subclass of von Neumann regular matrices consists of strongly regular matrices. In this direction, we also establish an intrinsic characterization of such an $n\times n$-matrix $A$ with $\rho(A)=$ over a commutative ring to be strongly regular, namely the trace of its compound matrix $C_t(A)$ to be invertible. In each case, we construct a (strong) inner inverse of our matrix. Finally, we derive applications to products of local commutative rings, and we count the matrices having these properties over some finite rings of residue classes and group algebras. The talk is based on recent joint work with Septimiu Crivei and Gabriela Olteanu.

BABES-BOLYAI UNIVERSITY - FACULTY OF MATHEMATICS AND COMPUTER SCIENCE
CLUJ-NAPOCA, CLUJ
iulia dot chiru at ubbcluj dot ro

On some classes of subalgebras of Leavitt path algebras

ANNA CICHOCKA, Warsaw University of Technology

The history of Leavitt path algebras dates back to the 1960s, when William G. Leavitt posed a question regarding the existence of $R$ rings that satisfy the equality $R^i=j$ as right-hand modules over R. The concept of Leavitt path algebras was introduced between 2005 and 2007. This construction was developed to algebraically represent combinatorial objects associated with Cuntz-Krieger algebras and $C^*$-algebras. The study of Leavitt path algebras is of interest to both algebraists and functional analysts, particularly those working with $C^*$-algebras. The flexible nature of Leavitt path algebra construction allows for the generation of numerous examples of algebras with specific properties.

An important special case of Leavitt path algebras is the class of matrix algebras $M_n(F)$ over a field F, where $n$ is a natural number or infinity (infinite-size matrices have a countable number of rows and columns with a finite number of non-zero elements in each). In my presentation, I will recall definition of Leavitt path algebras. Based on this, I will demonstrate a construction of maximal commutative subalgebras of Leavitt path algebras and their connections with well-known examples of maximal commutative subalgebras of matrix algebras over a field.

WARSAW UNIVERSITY OF TECHNOLOGY
WARSAW,
Anna dot Cichocka at pw dot edu dot pl

The Osofsky Theorem and purity-related results:

from modules to finitely accessible additive categories

SEPTIMIU CRIVEI, Babes-Bolyai University

The classical Osofsky Theorem characterizes semisimple rings as the rings $R$ for which every quotient of $R$ is an injective right $R$-module. On the other hand, Ikeda and Nakayama described von Neumann regular rings as the rings $R$ for which every quotient of $R$ is an absolutely pure right $R$-module. Also, Gómez Pardo and Guil Asensio showed that if $R$ is a pure-injective right $R$-module, then the ring $R$ is semiperfect if and only if every pure quotient of $R$ is pure-injective. We explore how these results can be extended to finitely accessible additive categories. We also give some applications to comodule categories.

BABES-BOLYAI UNIVERSITY
CLUJ-NAPOCA,
septimiu dot crivei at ubbcluj dot ro

On some classes of almost maximum distance separable constacyclic codes over finite fields

HAI DINH, Department of Mathematics, Kent State University

Almost maximum distance separable (AMDS) symbol-pair codes are getting a lot of attention as they are multipurpose instruments that provide reliable data transfer in satellite and wireless communication systems, safeguard data integrity in storage systems, improve cryptography security, and facilitate error correction in both wireless and satellite networks. In this talk we construct several classes of AMDS constacyclic codes with good symbol-pair distances over finite fields. Applications and open direction for future research will also be discussed.

KENT STATE UNIVERSITY
KENT, OHIO
hdinh at kent dot edu

When are Baer modules extending?

FATMA AZMY EBRAHIM, Al-Azhar University

The concept of an extending module is closely tied to that of a Baer module. A right $R$-module $M$ is called extending if every submodule of $M$ is essential in a direct summand. A right $R$-module $M$ is called Baer if for all $N\leq M$, $l_S (N ) \leq^\oplus {}_{S}S$, where $S =t{End}_R (M)$. In 2004, Rizvi and Roman generalized a result of Chatters and Khuri (1980) in terms of modules, demonstrating the interplay between Baer and extending modules through the result: "a module $M$ is $\mathcal{K}$-nonsingular extending if and only if $M$ is $\mathcal{K}$-cononsingular Baer”. Here, $M_R$ is $\mathcal{K}$-nonsingular if $\forall \varphi \in S$ such that Ker$\varphi \leq^e M$, $\varphi =$. Furthermore, $M_R$ is $\mathcal{K}$-cononsingular if for any $N\leq M$ with $\varphi N \neq 0$ for all $0 \neq \varphi \in S$, it implies $N\leq^e M$. Consequently, every Baer module that is $\mathcal{K}$-cononsingular becomes an extending module. This talk primarily focuses on investigating the $\mathcal{K}$-cononsingularity of modules. Our research is driven by the observation that there is limited understanding of $\mathcal{K}$-cononsingularity compared to other related notions. Additionally, we introduce the concept of special extending (or sp-extending) modules and establish that the class of $\mathcal{K}$-cononsingular modules encompasses both extending and special extending modules. Among other findings, we present a new analogous version of Rizvi and Roman's result, highlighting the inherent connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided.
Joint work with: S. Tariq Rizvi and Cosmin S. Roman

AL-AZHAR UNIVERSITY
CAIRO,
fatema_azmy at azhar dot edu dot eg

On pseudo-continuous hulls and $C4$-hulls of modules and rings.

HUSSEIN EID EL-SAWY, Department of Mathematics, Faculty of Science,

Cairo University

One of the most fundamental facts in the subject of injective modules is that every module has a unique (up to isomorphism) injective hull $E(M)$. Johnson and Wong proved that every module has a unique quasi-injective hull in its injective hull. Goel and Jain proved that every module has a unique quasi-continuous hull in its injective hull and that isomorphic modules have isomorphic quasi-continuous hulls. Later, Birkenmeier, Park, and Rizvi studied the existence of continuous hulls of modules and rings. Recall that a module $M_R$ is called $C4$ if whenever $M=A\oplus B$ and $f:A\to B$ is a homomorphism with $\ker f \subseteq^{\oplus} A$, then Im $f \subseteq^{\oplus} B$. A module $M$ is called pseudo-continuous if it is both CS and $C4$. In this talk, we discuss when a module or a ring has a pseudo-continuous hull of a $C4$-hull.

DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, CAIRO UNIVERSITY
CAIRO,
heid at sci dot cu dot edu dot eg

Semidirect products

ALBERTO FACCHINI, Universitá di Padova

First of all, I will recall the well known notion of semidirect product both for classical algebraic structures (like groups and rings) and for more recent ones (digroups, left skew braces, heaps, trusses). Then I will deal the concept of semidirect product for an arbitrary algebra $A$ in a variety $\cal{V}$ of type $\cal{F}$. Here algebra means in the sense of Universal Algebra. An inner semidirect-product decomposition $A=B\ltimes\omega$ of $A$ consists of a subalgebra $B$ of $A$ and a congruence $\omega$ on $A$ such that $B$ is a set of representatives of the congruence classes of $A$ modulo $\omega$. An outer semidirect product is the restriction to $B$ of a functor from a suitable category $\cal{C}_B$ containing $B$, called the enveloping category of $B$, to the category $Set_*$ of pointed sets.

UNIVERSITá DI PADOVA
PADOVA,
facchini at math dot unipd dot it

Descending chains of coprime pairs and the exchange property

PEDRO ANTONIO GUIL ASENSIO, University of Murcia

(abstract)

UNIVERSITY OF MURCIA
MURCIA, SPAIN
paguil at um dot es

Purely Essentially Baer Modules

ASHOK JI GUPTA, Indian Institute of Technology(BHU)

A module M is said to be a purely essentially Baer if the right annihilator in M of any left ideal of the endomorphism ring of M is essential in a pure submodule of M. We study some properties of purely essentially Baer modules and characterize von Neumann regular rings in terms of purely essentially Baer modules.

INDIAN INSTITUTE OF TECHNOLOGY(BHU)
VARANASI, UTTAR PRADESH
agupta dot apm at itbhu dot ac dot in

On the Lie triple superderivations of generalized matrix algebras

LEILA HEIDARI ZADEH, Department of Mathematics and Statistics,

Shoushtar Branch, Islamic Azad University, Shoushtar, Iran

In this paper, we first determine the structure of Lie triple superderivations of generalized matrix algebras. In particular, under some mild conditions we present the necessary and sufficient conditions for a Lie triple superderivation to be proper.

DEPARTMENT OF MATHEMATICS AND STATISTICS, SHOUSHTAR BRANCH, ISLAMIC AZAD UNIVERSITY, SHOUSHTAR, IRAN
SHOUSHTAR,
heidaryzadehleila at yahoo dot com

Monoids of infinitely generated projective modules

and applications to direct sums decompositions

DOLORS HERBERA, Universitat Autonoma de Barcelona

Centre de Recerca Matemática

We want to present some results on monoids on monoids associated to non-necessarily finitely generated projective modules, taking into account different sources in the recent literature appeared on the topic, as well as its applications into direct sum decompositions of modules over suitable classes of rings like one dimensional noetherian commutative rings and h-local domains.

UNIVERSITAT AUTONOMA DE BARCELONA/CENTRE DE RECERCA MATEMáTICA
BELLATERRA, BARCELONA
dolors dot herbera at uab dot cat

Counting ideal generators for $C([0,1])$

DANIEL HERDEN, Baylor University

We consider the ring $R=[0,1]$ of real-valued continuous functions on the compact interval $[0,1]$ with pointwise addition and multiplication of functions. It is well-known that the maximal ideals of $R$ are given by $I_a :=\{f\in R \mid f(a)=0\}$ for $a\in [0,1]$ but how many generators are required for the ideal $I_a$? We will demonstrate how the answer to this question relates to the dominating number $\mathfrak{d}$, a cardinal characteristic of the continuum.

BAYLOR UNIVERSITY
WACO, TX
daniel_herden at baylor dot edu

On Artinian Rings

DINH VAN HUYNH, Professor Emeritus Ohio University

In this talk, we will prove the following theorem: Let $R$ be a right and left Artinian ring, then R = A+B, where $A$ and $B$ are ideals with intersection zero, and $(A, +)$ has DCC on subgroups, every nonzero left or right ideal of B is infinite.

PROFESSOR EMERITUS OHIO UNIVERSITY
PALO ALTO, CALIFORNIA
Huynhohio at gmail dot com

Ideals on a Projective Model

AYCIN IPLIKCI ARODIRIK, The Ohio State University

We will focus on birational projective models, blow up of an ideal, and introduce Abhyankar's definition of 'ideal on a model'. We will extend the notion of homogeneous ideals of polynomial rings to Nagata function rings. Afterwards, as with the Proj- construction, we will present another way of seeing these "ideals" as the ideals of a larger ring.

THE OHIO STATE UNIVERSITY
COLUMBUS, OHIO
iplikciarodirik dot 1 at osu dot edu

Intermediate rings in extensions of commutative rings

with some finiteness conditions

ALI JABALLAH, University of Sharjah

We establish several finiteness characterizations and equations for the cardinality of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions.

UNIVERSITY OF SHARJAH
SHARJAH, SHARJAH
ajaballah at sharjah dot ac dot ae

Wreath Product of Algebras and Embedding Theorems

S.K. JAIN, Professor Emeritus Ohio University

The concept of wreath product of algebras had been first introduced in (2017). The first work on this topic appeared by, Alahmadi-Alsulami-Jain- Zelmanov, Trans AMS (2019), followed by papers which studied among others topics, the embeddability problems of countably generated algebras into finitely generated algebras. We will talk of some such embeddings that answer open questions.

PROFESSOR EMERITUS OHIO UNIVERSITY
ATHENS, OHIO
jain at ohio dot edu

Fixed product preserving maps on alternative division rings

HAYDEN JULIUS, Niagara University

In this talk, we characterize additive maps $f: \mathcal{A} \to \mathcal{A}$ on an alternative division ring $\mathcal{A}$ with the property that $f(x)f(y)=m$ whenever $xy=k$, where $k$ and $m$ are fixed, nonzero elements. Such a map is called a fixed product preserving map. We find $f$ is a so-called Jordan automorphism, up to multiplication on the left by a fixed element. The ideas are then extended to alternative Cayley-Dickson algebras of dimension $2, 4$ or $8$ over its center, including split algebras, when $m$ is invertible and the characteristic is not 2. Interestingly, Jordan automorphisms need not be automorphisms or antiautomorphisms unlike in the associative case. Some open problems for non-invertible $m$ and characteristic 2 algebras are presented. We also characterize additive maps $\phi: \mathcal{A} \to \mathcal{A}$ satisfying a weaker inverse preserving property given by $\phi(x)\phi(x^{-1})\phi(x) =hi(x)$ for all $x \in \mathcal{A}^\times$, and s how that $\phi(x^{-1}) =hi(x)^{-1}$ is automatic. This final property turns out to be essential for understanding any fixed product preserving map.

NIAGARA UNIVERSITY
NIAGARA FALLS, NEW YORK
hjulius at niagara dot edu

N-Koszul Algebras of Finite Global Dimension

ABDOURRAHMANE KABBAJ, University of California Irvine

The class of $N$-koszul graded algebras of finite global dimension has gained lots of attention in recent years, especially in the study of Artin-Schelter regular algebras. While structurally rich and concrete, the only known examples of such algebras are either when $N =2$, i.e. the algebra is Koszul, or when $N =3$. Under a mild Hilbert series assumption, we rule out the existence of $N$-koszul graded algebras of finite global dimension for $N$ not prime. Furthermore, we establish restrictions on the global dimension of such algebras.

UNIVERSITY OF CALIFORNIA IRVINE
IRVINE, CA
akabbaj at uci dot edu

The Role of Algebraic Structures in Certain Cryptosystems

ZEKERIYA YALCIN KARATAS, University of Cincinnati Blue Ash College

In the last few decades, cryptography has seen the rise of several novel cryptosystems based on various algebraic structures. These cryptosystems make it possible to securely encrypt, decrypt, and exchange keys, ensuring safe communication. Some of these methods have been around to be even cryptanalyzed. In this talk, we will present various cryptosystems based on some major algebraic structures. Some cryptanalysis examples will be included in the presentation as well. The co-presenter in this talk is Dr. Charles Emenaker from UC Blue Ash.

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

On the generalized convergence of sequences

KUMAR KAUSHIK, Chandigarh University

In the present talk, I would like to highlight some types of generalized convergence of sequences.

CHANDIGARH UNIVERSITY
MOHALI, PUNJAB
kaushikvjy at gmail dot com

The Schröder-Bernstein problem for dual F-Baer modules

DERYA KESKIN TÜTÜNCÜ, Hacettepe University

In module theory, the Schröder-Bernstein problem states that if $A$ and $B$ are two modules such that there is a monomorphism from $A$ to $B$ and a monomorphism from $B$ to $A$, then $A\cong B$.

Let $M$ be a module and $F$ a fully invariant submodule of $M$. $M$ is called dual $F$-Baer if for every family of homomorphisms $\{f_\alpha\}_{\alpha\in I}$ of $M$, $\sum_{\alpha \in I}g_\alpha(F)$ is a direct summand of $M$ (see [1] and [2]). In this talk we show that any module $M$ is dual $F$-Baer if and only if $M=\oplus N$ for some submodule $N$ of $M$ and $F$ is dual Baer (see [2, Theorem 2.3]). Remember that a module $M$ is called dual Baer if for every family of homomorphisms $\{f_\alpha\}_{\alpha\in I}$ of $M$, $\sum_{\alpha \in I}g_\alpha(M)$ is a direct summand of $M$ (see [3]).

Let $R$ be a ring and $r$ a preradical of Mod-$R$. Let $M$ and $N$ be two right $R$-modules. Assume that $M$ is dual $r(M)$-Baer and $N$ is dual $r(N)$-Baer such that $M=M)\oplus X$ and $N=N)\oplus Y$ and $X\cong Y$. Then we show that $M$ and $N$ satisfy the Schröder-Bernstein property (see [2, Theorem 2.7]). We illustrate our hypothesis with an example constructed over the ring of integers $\mathbb{Z}$ (see [2, Example 2.8]).

[1] S. Crivei, D. Keskin Tütüncü and G. Olteanu, $F$-Baer objects with respect to a fully invariant short exact sequence in abelian categories, Communications in Algebra 49, 5041-5060 (2021).

[2] D. Keskin Tütüncü, The Schröder-Bernstein problem for dual $F$-Baer modules, Journal of Algebra and Its Applications 21, 2250131 (6 pages) (2022).

[3] D. Keskin Tütüncü and R. Tribak, On dual Baer modules, Glasgow Math. Journal, 52, 261-269 (2010).

HACETTEPE UNIVERSITY
ANKARA,
keskin at hacettepe dot edu dot tr

Some Irreducibility Criteria for polynomials over rationals

SUDESH KAUR KHANDUJA, IISER Mohali

We will review the classical irreducibility criteria of Eisenstein, Sch $\ddot{\mbox{o}}$nemann and Dumas, and then discuss their extensions using Newton polygons and the theory of valuations. We shall also present recently proved generalizations of the well known result of Schur. This talk is partly based on joint work with Ankita Jindal.

IISER MOHALI
MOHALI,
skhanduja at iisermohali dot ac dot in

Leavitt Path Algebras of Small Gelfand-Kirillov Dimension

AYTEN KOÇ, Gebze Technical University

I will report on an ongoing joint project with Murad Özaydın concerning representations of Leavitt path algebras (LPAs) of polynomial growth. The ultimate goal is to classify LPAs of polynomial growth up to Morita equivalence. LPAs are associated to di(rected )graphs and there is a combinatorial procedure (the reduction algorithm) making the digraph smaller while preserving the Morita type. We can recover the vertices and most of the arrows of the completely reduced digraph from the module category of the LPA. As a corollary we obtain a complete Morita invariant (the weighted Hasse diagram) when the Gelfand-Kirillov dimension of the LPA is less than 4.

*Partially supported by TÜBİTAK ARDEB 1001 grant 122F414

GEBZE TECHNICAL UNIVERSITY
GEBZE, KOCAELI
ozgayten at gmail dot com

Duality in Derived Category O and Locally Analytic Representations

CEMILE KURKOGLU,

A representation of a $p$-adic reductive group $G$ leads to a representation of the associated Lie algebra $\mathfrak{g}=ext{Lie}(G)\,.$ In the other direction, going from $\mathfrak{g}$-modules to representations, S. Orlik and M. Strauch have exhibited a functor $\mathcal{F}$ from the BGG category $\mathcal{O}$ to the category of locally analytic representations, which was later generalized to the extension closure $\mathcal{O}^\infty\,.$ The derived category of category $\mathcal{O}^\infty$ carries an Ext-duality functor $\mathbb{D}^g\,.$ On the other hand, P. Schneider and J. Teitelbaum have defined and studied a duality functor $\mathbb{D}^G$ for locally analytic representations of $G\,.$ The main result of this talk explains how, under certain technical assumptions, the functor $\mathcal{F}$ interacts with the duality functors $\mathbb{D}^g$ and $\mathbb{D}^G\,.$

COLUMBUS, OHIO
cemile dot kurkoglu at gmail dot com

On the excellence for inseparable quartic extensions

AHMED LAGHRIBI, Artois University

A field extension $K/F$ is called excellent for quadratic forms if for any $F$-quadratic form $Q$, the anisotropic part of the $K$-quadratic form $Q_K$ is defined over $F$. When $K/F$ is of degree 2 or 3, it is known that $K/F$ is excellent for quadratic forms. In this talk, we discuss the excellence property when $K/F$ is an inseparable quartic extension. To this end we combine arguments on Kato's cohomology, division algebras and transfer. (This is a joint work with Diksha Mukhija.)

ARTOIS UNIVERSITY
LENS,
ahmed dot laghribi at univ-artois dot fr

Quasi-Baer ring and module hulls

GANGYONG LEE, Chungnam National University

Kaplansky introduced the notions of Baer and Rickart rings in [1]. He and many others obtained a number of interesting results on these two classes of rings which have their roots in Functional Analysis. A ring $R$ is called a Baer (Rickart) ring if the right annihilator of any nonempty subset (singleton element) of $R$ is given by $eR$ with $e^2=e\in R$. Also, it has been of interest to investigate finite dimensional algebras over an arbitrary algebraically closed field. Clark [2] initially defined a quasi-Baer ring $R$ (i.e., a ring $R$ for which the left annihilator of every ideal is given by $Re$ with $e^2=e\in R$) to help characterize a finite dimensional algebra over an algebraically closed field to be a twisted semigroup algebra, which is also a generalization of Baer rings.

More recently, the notions of a (quasi-)Baer was extended to analogous module theoretic notions using the endomorphism ring of a module by Rizvi and Roman in 2004. A module $M$ is said to be (quasi-)Baer if the right annihilator in $M$ of any subset (two-sided ideal) of End$_R(M)$ is generated by an idempotent in End$_R(M)$. A number of interesting papers have been published on these concepts in recent years.

In this talk, we introduce the definition and some results of quasi-Baer ring and module (hulls). As one application of quasi-Baer ring to $C^*$-algebra, we introduce the result that for a unitary $C^*$-algebra $A$, $A$ is boundedly centrally closed if and only if $A$ is a quasi-Baer ring. It is also one of applications of quasi-Baer ring hull. Also we proved 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 module hull. For a given module $M$, the smallest quasi-Baer overmodule of $M$ in $E(M)$(the injective module of $M$) is called the quasi-Baer module hull of $M$.

(This work is a joint work with Jae Keol Park, S. Tariq Rizvi, and Cosmin S. Roman)

[1] I. Kaplansky, Rings of operators, W. A. Benjamin, Inc., New York-Amsterdam (1968)

[2] W.E. Clark, Twisted matrix units semigroup algebras, Duke Math. J., 1967 34, 417–423

CHUNGNAM NATIONAL UNIVERSITY
DAEJEON,
lgy999 at cnu dot ac dot kr

Singular elements as product of idempotents

ANDRÉ GÉRARD LEROY, Université d'Artois

Starting with Howie and Erdös the problem of writing a singular matrix (or more generally a singular element of a ring) into a product of idempotents is related to some important questions. This is the case in particular for von Neumann regular rings. We will first give an overview of the main results in this area and then concentrate on von Neumann regular rings. If time permits we will also have a look at closely related problems.

UNIVERSITé D'ARTOIS
LENS, HAUTS DE FRANCE
andre dot leroy at univ-artois dot fr

Injective modules over Noetherian rings

CHRISTIAN EDGAR LOMP, University of Porto

Matlis showed that injective hulls of simple modules over commutative Noetherian rings are Artinian. In this talk, I will survey to what extent Matlis' result (or a weaker version of it) generalizes to non-commutative Noetherian rings and will review the works of Ken Brown, Paula Carvalho, Can Hatipoglu, Jurek Matczuk, Ian Musson, and myself.

UNIVERSITY OF PORTO
PORTO,
clomp at fc dot up dot pt

Polynomial Dimension of a Ring

K(ENNETH) ALAN LOPER, Ohio State University

We introduce a new type of dimension for the dimension of a ring which is much better behaved than Krull dimension and various other dimensions.

OHIO STATE UNIVERSITY
COLUMBU, OHIO
loper dot 4 at osu dot edu

Some Remarks on S-Noetherian Modules

SANJEEV KUMAR MAURYA, Department of Mathematics, SBS, Galgotias University

In this paper, we study several properties and applications of S-Noetherian rings and modules. We prove S-version of Akizuki's theorem. Furthermore, we prove that associated primes exist in modules over S-Noetherian rings and most of the notions of associated prime ideals coincide over S-Noetherian rings. We also extend classical Krull's intersection theorem for S-Noetherian rings.

DEPARTMENT OF MATHEMATICS, SBS, GALGOTIAS UNIVERSITY
GREATER NOIDA, UTTAR PRADESH, INDIA
sanjeevm50 at gmail dot com

Simple and amenable bases in polynomial algebras (preliminary report

of a collaboration with Fatma Ebrahim and Sergio R. Lopez-Permouth)

AZAM MOZAFFARIKHAH, ohio university

We report on three accomplishments obtained during our recent work. The questions we address are:
$1.$ Whether the bases of monomials constitute simple bases for algebras of polynomials,
$2.$ Searching for a complete (up to mutual congeniality) list of simple bases for the algebra of polynomials in a single variable.
Our results include: a) We show that the bases consisting of monomials are always simple for the corresponding algebra. b) When the family of all polynomials on a set $X$of variables is considered as a module over the algebra of polynomials over a proper subset $Y$ of $X$, the monomial bases are not simple. This strengthens the only previously known result when $X$ is a singleton ([1]). c) For any value of $\beta$in the field, the collection of powers of the polynomial $x+\beta$ is a basis. Such basis, denoted $P_{\beta}$, was named the Pascal basis induced by $x+\beta$ in [1], where it was shown that the family of Pascal bases consists of discordant simple bases. We extend the list of simple bases for the algebra of single variable polynomials by considering a new family of bases that we will call hyper-Pascal bases, obtained via a suitable modification of the construction of the Pascal bases.
Keywords: Amenable bases; congeniality of bases; proper congeniality; simple bases; infinite dimensional algebras; algebra of polynomials
References:
$[1]$ Al-Essa, López-Permouth and Muthana, Modules over infinite dimensional algebras, Linear and Multilinear Algebra 66 (3), 488-496 (2018).
$[2]$ P. Aydogdu, C. Arellano, S.R. López-Permouth, R. Muhammad, and M. Zailaee, The isomorphism problem for basic modules and the divisibility profile of the algebra of polynomials, submitted for consideration.

OHIO UNIVERSITY
ATHENS, OH
mozaffarikhah at ohio dot edu

Group collaborations and braces (preliminary report)

AARON NICELY, Ohio University

Following a recent Semigroup Forum paper by the speaker with S. Lopez-Permouth and M. Zailaee, given two operations $\ast$ and $\circ$ on a set $S$, an operation $\star$ on $S$ is said to be a collaboration between $\ast$ and $\circ$ if for all $a,b \in S$, $a \star b$ $\in \{a \ast b, a\circ b \}$. Given a not-necessarily commutative group $(G,+)$, we define the operation $-$ by $a-b=(-b)$. We are interested in learning what collaborations of $+$ and $-$ are such that $(G, \ast)$ is group. A group is said to be collaborative if it has nontrivial collaborations (i.e a collaboration $\ast \ne +$. We will explore basic facts about collaborative groups and see examples and counterexamples of collaborative finite groups of small order. We will consider when, for a collaborative group $(G,+)$ with nontrivial collaboration $\ast$, $(G,+,\ast)$ is a skew left brace. This is a preliminary report on joint work with Sergio Lopez-Permouth.

OHIO UNIVERSITY
ATHENS, OHIO
an636517 at ohio dot edu

Separativity and its connections to multiple ring-theoretic conditions

PACE P. NIELSEN, Brigham Young University

There are many cancellation conditions for direct sum decompositions in module categories. In the late 1990s, separativity was introduced as another important cancellation property, especially in the context of (von Neumann) regular rings and exchange rings. There were some immediate connections to important ring-theoretic properties, like stable range conditions, unit-regularity, Hermiteness (the ability to diagonally reduce matrices), writing invertible transformations as products of elementary operations and diagonal matrices, and other properties.

These connections are outlined and explored, and a few new ones are introduced along the way. For instance, it is explained how separative regular rings can be viewed as a subvariety of the regular rings. These same ideas apply to other cancellation properties, giving them similar varietal descriptions. As another example, the exact minimal number of elementary operations in separative regular rings, needed to take a generic $2\times 2$ matrix to a diagonal matrix, is described.

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

Kernel and Co-Kernel decomposition of the Magma Monoid

ISAAC OWUSU-MENSAH, Akenten Appiah-Menka University

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 and terminology $\mathcal {M}(S)$ (the magma of $S$) to denote the set of all binary operations on the set $S$ (i.e. the set of all magmas with underlying set $S$.)

Our study concerns a monoid structure $(\mathcal{M}(S), \triangleleft )$ satisfying that each outset, $out(\ast)= \circ \in \mathcal{M}(S)\vert \ast$$\text { distributes over } \circ \}$, is a submonoid. This endowment gives us a possibility to compare the properties of an operation $\ast \in \mathcal {M}(S)$ and those of the monoid structure of $(out(\ast), \triangleleft)$. We determine that isomorphic operations yield isomorphic outsets and explore possible converses for that result.

In this presentation, we will obtain a generic decomposition, called the kernel-cokernel decomposition, of arbitrary magmas and ideals. We also characterize those cases when a cokernel-kernel decomposition is also possible as we introduce anticommutative and pseudo-anticommutative operations. ( A collaborative work with Sergio Lopez-Permouth and Asiyeh Rafeiepour)

AKENTEN APPIAH-MENKA UNIVERSITY
MAMPONG-ASHANTI, ASHANTI
iowusumensah at aamusted dot edu dot gh

On the Big lattices of hereditary and natural classes

of linear modular lattices.

SEBASTIAN PARDO-GUERRA, UCSD

The collection of hereditary classes of modules over an arbitrary ring $R$ is a pseudocomplemented complete big lattice. The elements of its skeleton are precisely the natural classes of $R$-modules. In this talk, I present the correspoinding results about hereditary classes in $R$-Mod to the category $\mathcal{L_{M}}$ of linear modular lattices, which has as objects all complete modular lattices, and as morphisms all linear morphisms. Also, natural classes are defined in the full subcategory $\mathcal{L_{M}}_{c}$ of upper semicontinuous modular complete lattices; these classes comprise the skeleton of the big lattice of hereditary classes in $\mathcal{L_{M}}_{c}$.

UCSD
LA JOLLA, CALIFORNIA
spardoguerra at ucsd dot edu

Some aspect on various generalizations of $h-$ Lifting Modules

MANOJ KUMAR PATEL, National Institute of Technology Nagaland

In this talk, we introduce and study the properties of $hw-$lifting modules as a special extensions of $h-$lifting which are further generalization of lifting modules. We observed that finite direct sum of $hw-$lifting modules may not be $hw-$lifting, so we provided several sufficient conditions for which $hw-$lifting modules are inherited under direct sum. Moreover, we have introduced and studied the properties of other versions of $h-$lifting module namely; $mh-$lifting, $mhw-$lifting and completely $mh-$lifting modules and established several properties of the $hw-$lifting modules in terms of these modules.

Keywords: Lifting module, hollow module, coessential submodule, maximal semisimple submodule.

Bibliography.

[1] D. Keskin, On lifting modules, Communications in Algebra, 28(7)(2000) 3427-3440.

[2] D. Keskin, A. Harmanci, A relative version of the lifting property of modules, Algebra Colloquium, 11(3)(2004) 361-370.

[3] K. Oshiro, Lifting modules, extending modules and their applications to generalized uniserial rings, Hokkaido Mathematical Journal, 13(3)(1984) 339-346.

[4] M. K. Patel, S. K. Choubey and L. K. Das, On finitely-hollow-weak lifting modules, Proceedings of the Indian National Science Academy, 87(1)(2021) 143-147.

[5] Y. Wang and D. Wu, Direct sums of hollow-lifting modules, Algebra Colloquium, 19(1)(2012) 87-97.

NATIONAL INSTITUTE OF TECHNOLOGY NAGALAND
DIMAPUR, NAGALAND
mkpitb at gmail dot com

Nontriviality of rings of integral-valued polynomials

GIULIO PERUGINELLI, University of Padova

In 2012, Alan Loper and Nicholas J. Werner introduced the family of rings of integral-valued polynomials over subsets $S$ of the ring of all algebraic integers $\overline{\mathbb{Z}}$, defined as Int$_{\mathbb{Q}}(S,\overline{\mathbb{Z}})=f\in\mathbb{Q}[X]\mid f(S)\subseteq\overline{\mathbb{Z}}\}$, in order to obtain an example of a Prüfer domain strictly contained between $\mathbb{Z}[X]$ and the classical ring of integer-valued polynomials Int$(\mathbb{Z})$. We correct here a minor wrong claim of that paper, namely, that for any $S\subseteq\overline{\mathbb{Z}}$, the ring Int$_{\mathbb{Q}}(S,\overline{\mathbb{Z}})$ is always nontrivial, i.e., it strictly contains $\mathbb{Z}[X]$. For example, if $S$ comprises all the roots of unity $\xi_n,n\in\mathbb{N}$ it is not difficult to show that Int$_{\mathbb{Q}}(S,\overline{\mathbb{Z}})=\mathbb{Z}[X]$ using the fact that the ring of integers of $\mathbb{Q}(\xi_n)$ is monogenic. We completely characterize those subset s $S$ of $\overline{\mathbb{Z}}$ for which Int$_{\mathbb{Q}}(S,\overline{\mathbb{Z}})$ is nontrivial in terms of pseudo-divergent sequences and pseudo-stationary sequences contained in $S$ with respect to some fixed extension of the $p$-adic valuation to $\overline{\mathbb{Q}}$, as $p$ runs through the set of prime integers. These sequences have been introduced by Chabert in 2010 in order to study the polynomial closure of subsets of rank one valuation domains.

We produce several examples of subsets of $\overline{\mathbb{Z}}$ of unbounded degree for which the corresponding ring Int$_{\mathbb{Q}}(S,\overline{\mathbb{Z}})$ is trivial or not. In particular, we show that the monogenicity of the ring of integers of $\mathbb{Q}(s)$, for $s\in S$, is not a necessary condition for Int$_{\mathbb{Q}}(S,\overline{\mathbb{Z}})$ to be trivial.

This is a joint work with Nicholas J. Werner.

References

• G. Peruginelli, N. J. Werner, Nontriviality of rings of integral-valued polynomials, preprint, 2024.
• K. Loper, N. J. Werner. Generalized rings of integer-valued polynomials, J. Number Theory 132 (2012), no. 11, 2481-2490.

UNIVERSITY OF PADOVA
PADOVA,
gperugin at math dot unipd dot it

On Naimark's Problem for Leavitt Path Algebras

KULUMANI M. RANGASWAMY, University of Colorado, Colorado Springs

Naimark's original problem asked whether a C*-algebra possessing exactly one irreducible representation up to isomorphism characterizes the algebra of compact operators on a Hilbert space. Recently, this problem was solved in the affirmative for graph C*-algebras. In this talk, we consider Naimark's problem in the context of Leavitt path algebras, which are algebraic analogs of C*-algebras. Leavitt path algebras $L_K(E)$ of an arbitrary graph $E$ over a field $K$ possessing exactly one isomorphism class of simple left/right modules are characterized algebraically and graphically. A corresponding graded version of Naimark's problem is also solved. Naimark's problem for arbitrary associative algebras over a field is discussed.

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

Fundamental units of real quadratic number fields

and a conjecture of Mordell

ANDREAS REINHART, University of Graz /

Department of Mathematics and Scientific Computing

Let $d\geq 2$ be a squarefree integer, let $K=\mathbb{Q}(\sqrt{d})$ be the (unique) real quadratic number field defined by $d$ and let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. For each positive integer $f$, let $\mathcal{O}_f=\mathbb{Z}+f\mathcal{O}_K$ be the unique order in $K$ with conductor $f$. Let $\varepsilon>1$ be the (unique) fundamental unit of $\mathcal{O}_K$.

There is a multitude of conjectures and results in the literature that are connected to the squarefree integers $d$ with $\varepsilon\in\mathcal{O}_d$. One of these conjectures is the Pellian equation conjecture of Mordell (that was formulated by several authors, including Mordell, between 1959 and 1961), which states that if $d$ is a prime number with $d\equiv 3\mod 4$, then $\varepsilon\not\in\mathcal{O}_d$. In this talk, we present and discuss a counterexample to Mordell's Pellian equation conjecture. Furthermore, we elaborate the relations of the condition “ $\varepsilon\in\mathcal{O}_d$” to some problems in number theory, factorization theory and module theory.

UNIVERSITY OF GRAZ / DEPARTMENT OF MATHEMATICS AND SCIENTIFIC COMPUTING
GRAZ, STYRIA
andreas dot reinhart at uni-graz dot at

When is Koszul algebra a domain?

MANUEL L. REYES, UC Irvine

Based on classical results in commutative algebra, it is natural to expect that a connected graded algebra or local ring with good homological properties should have no zero-divisors. However, such theorems are often difficult to prove for noncomutative rings unless one assumes very strong hypotheses. In this talk we will discuss a homological condition, stated in terms of the Ext algebra, that is necessary and sufficient for a Koszul algebra to be a domain. Similar methods yield a sufficient homological condition for a noncommutative local ring to be a domain and for a Koszul ring to be prime.

UC IRVINE
IRVINE, CA
mreyes57 at uci dot edu

Nonlinear Reed-Solomon codes

JESUS INDALECIO RUIZ BOLANOS, Baylor University

In error-correction codes, we are interested in designing strategies to detect and correct errors that arise from communication and storage problems. The ring of skew polynomials is a non-commutative collection of polynomials with very different properties when compared with the usual ring of polynomials. This structure has motivated more than one strategy to solve the error correction problem. In this work, we analyze a variety of $\mathbb{F}_q$-linear evaluation codes over $\mathbb{F}_{q^m}$, where the evaluation is realized in the classical sense. These codes exhibit properties similar to those found in the skew cyclic and skew quasi-cyclic codes introduced by Boucher & Ulmer and Abualrub et. al, respectively.

BAYLOR UNIVERSITY
WACO, TX
indalecio_ruizbolan1 at baylor dot edu

Prerradicals and idioms in abelian categories

MARTHA LIZBETH SHAID SANDOVAL-MIRANDA,

Universidad Autónoma Metropolitana (UAM), Campus Iztapalapa

A complete, modular and upper-continuous lattice is said to be an "idiom". Lattices with these characteristics were studied by H. Simmons in his research on theory of (pre)nuclei and generalizations on ring theory. Given an abelian Grothendieck category, it is well-known that the collection of subobjects of a given object happens to be an idiom. From this perspective, we can note that in the case of the categories of modules on a unitary ring; and more generally, for an abelian Grothendieck category, the theory of lattices and category theory is deeply connected.

Result of the research initiated in Mexico by F. Raggi, J. Ríos, R. Fernández-Alonso, H. Rincón and C. Singnoret, the study of preradicals has allowed us to obtain interesting results in theory of rings and modules through $R-pr$, the ”big idiom” of preradicals over a category of modules. In recent years, when studying the category of modules on a given ring, there have been adding the techniques from the pointfree-topology approach.

In this talk we will mention some results obtained with Ángel Zaldívar (CUCEI UdG) and Mauricio Medina (CIDESI) in the study of applications of point-free topology to module idioms; as well as some advances made together with R. Fernández-Alonso (UAM-Iztapalapa), J. Magaña Zapata (UAM-Azcapotzalco) and V. Santiago-Vargas (F. Ciencias UNAM), in generalizations of the study of preradicals in abelian categories.

Research supported by the Project "Interactions between topology, algebra and categories" (Programa de Proyectos de Investigación por Personal Académico de Ingreso Reciente, UAM 2024).

UNIVERSIDAD AUTóNOMA METROPOLITANA (UAM), CAMPUS IZTAPALAPA
MEXICO CITY , MEXICO
marlisha at xanum dot uam dot mx

Lattice of injective profile of rings and its impact on the ring structure

BULENT SARAC, Hacettepe University

Recent work has shown that the injective profile of a ring may happen to be a useful structure while investigating properties of the ring, and there has been a growing research interest in examining rings by imposing restrictions on their profile. Interestingly, in almost every work in which the profile of the ring is restricted, we see that the ring can be decomposed into a direct sum, with one of the two components being a semisimple Artinian ring and the other being an indecomposable ring having homogeneous right socle and in most cases this phenomenon is obtained as a result of very deep and comprehensive examinations. The aim of this work is to shed some light on this mystery and try to explain why this phenomenon occurs each time we consider the same type of problem.

HACETTEPE UNIVERSITY
ÇANKAYA, ANKARA
bsarac at hacettepe dot edu dot tr

Invariant subspaces of nilpotent linear operators: Algebra & Applications

MARKUS SCHMIDMEIER, Florida Atlantic University

Linear operators are ubiquitous in mathematics and its applications, we briefly discuss two situations, in control theory and in topological data analysis, where systems of invariant subspaces appear prominently.

How does my talk fit in a ring theory conference? Given a field $k$ and a natural number $n$, the bounded polynomial ring $\Lambda=k[T]/(T^n)$ is a self-injective algebra, hence the triangular matrix ring $U_2(\Lambda)$ is 1-Gorenstein. The Gorenstein-projective representations form the category $\mathcal S(n)$ of monic maps between $\Lambda$-modules: The objects are the pairs $X =(U,V)$, where $V$ is a finite-dimensional vector space with a nilpotent operator $T$ with $T^n =0$, and $U$ is a subspace of $V$ such that $T(U) \subseteq U$.

Our main goal is to discuss new numerical invariants for an object $X\in\mathcal S(n)$, the (Jordan-) level $pX$, the mean $qX$ and the colevel $rX$. The support $(pX,rX)$ of $X$ is a point in the triangle $\mathbb{T}(n)$. We will identify regions in $\mathbb{T}(n)$ in which indecomposable representations are either sparse or else occur in dense Brauer-Thrall families.

My presentation is a report about recent joint work with Claus Michael Ringel in Bielefeld.

FLORIDA ATLANTIC UNIVERSITY
BOCA RATON, FLORIDA
markusschmidmeier at gmail dot com

Graded automorphisms and twist of Leavitt path algebras

ASHISH SRIVASTAVA, Saint Louis University

In this talk we will present a general construction for graded automorphisms of Leavitt path algebras and as a consequence we will study the Zhang twist of Leavitt path algebras.

SAINT LOUIS UNIVERSITY
ST LOUIS , MO
ashish dot srivastava at slu dot edu

On the Extent of Weak-injectivity of Direct Sums of Modules

SULTAN EYLEM TOKSOY, Hacettepe University

Injective modules are among the most important homological objects in module categories. They play a significant role in both Module and Ring Theory, as well as in Homological Algebra. Determining whether a module is injective can be quite challenging. A new trend in Module Theory expands the study of module properties by developing mechanisms that not only determine when a property is exactly satisfied but also measure the extent to which a particular module fails to satisfy that property. This approach allows for the consideration of not only modules that satisfy a property, but also those that partially or even minimally satisfy it. Recently, the concept of the injectivity domain has gained increasing interest as an idea to measure the injectivity of a module. While not the only mechanism to measure the injectivity of a specific module, the injectivity domain is a good mechanism. The measuring tools for modules in this mechanism are portfolios. If a class of modules is an injectiv ity domain of a module, that class is called an (injective) portfolio, and the class of all possible (injective) portfolios for all right R-modules over a ring $R$ is called the (injective) profile of $R$. In [2], two different weak-injectivity domains used to measure the scope of injectivity or weak-injectivity were defined. In [4], a different perspective, which has not been used before, was aimed to contribute to the literature. It is well-known that the injectivity of certain modules determines the ring as Noetherian (see [3]). López-Permouth and Saraç showed in [4] that this criterion is not absolute but, instead, the injectivity domains of these specific modules serve to measure the extent to which a ring is Noetherian. In [1], it was shown that a ring is a q.f.d. ring if its certain modules are weakly-injective. In this work (small) weak-injectivity domains of modules are used to measure how q.f.d. a ring is. On the other hand in [4] López-Permouth and Saraç considered portfolios in the injective profile of a ring $R$ as layers and if the injectivity domain of the direct sum of every module family whose injectivity domain is a portfolio $\mathcal{A}$ is also $\mathcal{A}$, $\mathcal{A}$ is called a stable portfolio. They asked the question: is there an injective portfolio in the middle of the injective profile of the ring $R$, where every layer below it becomes stable? The answer to this question is yes, and this injective portfolio is called the Noetherian threshold. Based on these ideas, in this work we explore whether there is a (small) weak-injective portfolio in the middle of the (small) weak-injective profile of a ring $R$, where every layer below it is stable. Additionally, rings in which only poor modules have stable injective portfolios are called volatile rings in [4]. Also volatile rings are defined in conjunction with weak-inje ctivity, their properties are investigated, and examples are provided. Furthermore, the volatility of a ring using the weak-injective profile is examined.

[1] Al-Huzali, A. H, Jain, S. K., López-Permouth, S. R. 1992. Rings whose cyclics have finite Goldie dimension, J. Algebra, 153, 37-40.

[2] Aydoğdu, P., López-Permouth, S. R., Sandoval-Miranda, M. L. S. 2021. On the Weak-Injectivity Profile of a Ring, Bull. Malays. Math. Sci. Soc., 44, 35-53.

[3] Faith, C., Walker, E. A. 1962. Direct-sum representations of injective modules, Transactions of the American Mathematical Society, 103(3), 509-535.

[4] López-Permouth, S. R., Saraç, B. 2022. On the extent of the injectivity of direct sums of modules, Quaestiones Mathematicae, 1-12.

HACETTEPE UNIVERSITY
ANKARA,
eylemtoksoy @ hacettepe dot edu dot tr

Internal Coproduct, Coprime Modules and their Associated Topology

JESÚS VILLAGÓMEZ, Instituto de Matemáticas, UNAM

For $L$ and $N$ fully invariant submodules of an $R$-module $M$, the internal coproduct is defined in preradical terms as: $(L:_{M}N)=omega^{M}_{L}:\omega^{M}_{N})(M)$. We define the concept of coidempotent submodule and we obtain a new characterization of the semisimple rings. The concepts of coprime and semicoprime submodules (ideals) are defined. We study the properties of these modules and obtain results for Kasch modules (left Kasch rings). Finally we give a topology to the set of coprime submodules of an R-module M and provide a characterization when this topology is $T_{1}$ (Frechet).

INSTITUTO DE MATEMáTICAS, UNAM
MEXICO CITY,
jesusvc9197 at gmail dot com

MDS symbol-pair constacyclic codes of length $2^s$ over $\mathbf{F_{2^m}}.$

MANH THANG VO, Ohio University

In 2012, Chee et al. introduced the Singleton Bound for symbol-pair codes as follows: For any symbol-pair code $C$ of length $n$ over $\mathbb{F}_{2^m}$ with symbol-pair distance $s(C)$ such that $2 \le s(C) \le n$, $\vert C\vert\leq 2^{m(n-s(C)+2)}$. A symbol-pair code $C$ is called a maximum distance separable (briefly, MDS) symbol-pair code if it attains the Singleton Bound for symbol-pair codes, i.e., $\vert C\vert={m(n-s(C)+2)}$. MDS symbol-pair codes form an optimal class of symbol-pair codes as they have the best possible error-correction capability.

By using the determination of symbol-pair constacyclic codes we identify all MDS symbol-pair constacyclic codes of length $2^s$ over $\mathbb{F}_{2^m}$.

OHIO UNIVERSITY
ATHENS, OH
mv294721 at ohio dot edu

A Generalization of Zariski Topology

EDA YıLDıZ, Yildiz Technical University

Let $R$ be a commutative ring with nonzero identity and, $S\subseteq R$ be a multiplicatively closed subset. An ideal $P$ of $R$ with $P\cap S=mptyset$ is called an $S$-prime ideal if there exists an (fixed) $s\in S$ and whenever $ab\in P$ for $a,b\in R$ then either $sa\in P$ or $sb\in P$. In this talk, we talk about a topology on the set $Spec_{S}(R)$ of all $S$-prime ideals of $R$ which is generalization of prime spectrum of $R.$ We observe which of the separation axioms this topology satisfies. Moreover, we investigate the relations between algebraic properties of $R$ and topological properties of $Spec_{S}(R)$ like connectedness and irreducibility.

YİLDİZ TECHNICAL UNIVERSITY
ISTANBUL,
edyildiz at yildiz dot edu dot tr

Symmetry Questions about Amenability and Simplicity of Bases

MAJED HUSAIN ZAILAEE, King Abdulaziz University

We show that the properties of left amenability and left simplicity of bases are largely independent from their right-sided counterparts. Graph magma algebras are a suitable non-commutative setting for these considerations. This paper focuses on characterizing left and right amenability and simplicity of the so-called bases of vertices (or permissible bases) of such algebras. It is shown, among other results, that finite support magma algebras have no simple left or right bases.

An algebra $\mathcal{A}$ will be called right duo-amenable (or simply right duo when the meaning is clear from the context) if every right amenable basis of $\mathcal{A}$ is also left amenable. Connections between the properties of right duo and left duo are explored. This is a joint work with Fatma Azmy and Sergio López-Permouth

KING ABDULAZIZ UNIVERSITY
JEDDAH, SAUDI ARABIA
mzailaee at kau dot edu dot sa

Taking the Ext algebra is a process

that distributes across twisted tensor product

MAUREEN ZHANG, UC Irvine

Given any associative k-algebra A, the Ext algebra is a graded k-vector space with algebra structure given by the Yoneda product. The Ext algebra served as an important homological invariant for various reasons. Moreover, it is known that taking the Ext algebra is a process that distributes nicely across tensor product, i.e. $Ext(A \otimes B)\cong Ext(A) \otimes Ext(B)$. Therefore it is natural to ask if the same statement holds for the non-commutative analogue of tensor product, twisted tensor product.

This talk will give a positive answer in the case where A and B are Koszul, and $\tau$ is a strongly graded invertible twist.

UC IRVINE
IRVINE, CA
yuanfz1 at uci dot edu

Unit-fusible property via unit-regularity

YIQIANG ZHOU, Memorial University of Newfoundland

This is a preliminary report based on a joint work with P. Bhattacharjee and W.Wm. McGovern.

An element in a ring (associative with identity) is called left unit-fusible if it is the sum of a left zero-divisor and a unit, and the ring is left unit-fusible if each of its nonzero elements is left unit-fusible. These notions rose in a paper by Ghashghaei and McGovern [E. Ghashghaei and W.W. McGovern, Fusible rings. Comm. Algebra 45(3) (2017), 1151-1165], where, among others, it is shown that unit-regularity implies the unit-fusible property. In this talk, we present results on the question: Which rings satisfy the condition that every left unit-fusible element is unit-regular?

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