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We show that there is a little mistake in an implication in a paper of Bob Gilmer on rngs. [ABSTRACT FROM AUTHOR]

Zero-divisor graphs of a commutative ring R, denoted G(R), are well-represented in the literature. In this paper, we consider domination numbers of zero-divisor graphs. For reduced rings, Vatandoost and Ramezani characterized the possible graphs for G(R) when the sum of the domination numbers of G(R) and the complement of G(R) is n - 1, n, and n + 1, where n is the number of nonzero zero-divisors of R. We extend their results to nonreduced rings, determine which graphs are realizable as zero-divisor graphs, and provide the rings that yield these graphs. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [ABSTRACT FROM AUTHOR]

We fill a gap in the proof of one of the central theorems in Epp’s paper, concerning p-cyclic extensions of complete discrete valuation rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. Denote by modR the category of finitely generated R-modules. In the present paper, we introduce the notion of solid subcategories of modR and investigate it. The main result of this paper not only recovers results of Schoutens, Krause and Stevenson, and Takahashi on thick subcategories, but also unifies and extends them to solid subcategories. Moreover, it provides some contributions to the study of the question asking when a thick subcategory is Serre. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring R having a hyperideal I, the I-based zero-divisor graph Γ(I)(R) associated with R is the simple graph whose vertices are the elements of R∖I having their hyperproduct in I, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with I. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the I-based zero-divisor graph associated to the general hyperring Zn of the integers modulo n, when n=2pmq, with p and q two different odd primes, and fixing the hyperideal I. [ABSTRACT FROM AUTHOR]

The paper deals with new specific constructions of indecomposable torsion-free abelian groups of rank two and nonzero rings on them. They illustrate purely theoretical results and complement quite rare examples obtained during the classical as well as recent research of additive groups of rings. The presented results concerning the homogeneous groups remain true for groups of any finite nonzero rank. Moreover, the paper contains a construction of a torsion-free indecomposable abelian group of an arbitrary finite rank greater than two supporting an associative, but not commutative ring, as well as a ring which is neither associative nor commutative. [ABSTRACT FROM AUTHOR]

The SFT (for strong finite type) condition was introduced by [J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973) 299–304] in the context of studying the condition for formal power series rings to have finite Krull dimension. In the context of commutative rings, the SFT property is a near-Noetherian property that is necessary for a ring of formal power series to have finite Krull dimension behavior. Many others have studied this condition in the context of the dimension of formal power series rings. In this paper, we explore a specialization (and in some sense a more natural) variant of the SFT property that we dub as the VSFT (for very strong finite type) property. As is true of the SFT property, the VSFT property is a property of an ideal that may be extended to a global property of a commutative ring with identity. Any ideal (respectively, ring) that has the VSFT property has the SFT property. In this paper, we explore the interplay of the SFT property and the VSFT property. [ABSTRACT FROM AUTHOR]

The paper contains the classification of non-commutative torsion-free associative rings of rank two. Furthermore, torsion-free abelian groups of rank two supporting an associative, but not commutative ring structure are classified. [ABSTRACT FROM AUTHOR]

Let A be a commutative ring. In this paper, we introduce and study the concept of 2J-submodule A submodule U of a A-module Tis called a 2J-submodule) if for all a ∈ A and t ∈ T whenever at ∈ U with a² ∉ (j(A)T:T), then t ∈ U In this work examine the characteristics of the 2J-submodule as a generalization of the J-submodule. This paper provides various characterizations and properties of 2j-submodule. [ABSTRACT FROM AUTHOR]

The idempotent divisor graph of a commutative ring R is a graph with a vertex set in R* = R-{0}, where any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e^2 = e ∈ R, and is denoted by Л(R). Our goal in this work is to transform the planar idempotent divisor graph after coloring its regions into optical art by depending on the reflection of vertices, edges, and planes on the x or y-axes. That is, we achieve Op art solely through pure mathematics in this paper. [ABSTRACT FROM AUTHOR]

For a finite simple undirected graph G , the universal adjacency matrix U (G) is a linear combination of the adjacency matrix A (G) , the degree diagonal matrix D (G) , the identity matrix I and the all-ones matrix J , that is U (G) = α A (G) + β D (G) + γ I + η J , where α , β , γ , η ∈ ℝ and α ≠ 0. The cozero-divisor graph Γ ′ (R) of a finite commutative ring R with unity is a simple undirected graph with the set of all nonzero nonunits of R as vertices and two vertices x and y are adjacent if and only if x ∉ y R and y ∉ x R. In this paper, we study structural properties of Γ ′ (R) by defining an equivalence relation on its vertex set in terms of principal ideals of the ring R. Then we obtain the universal adjacency eigenpairs of Γ ′ (R) and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of Γ ′ (R) and Γ ′ (R) ¯ in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo m in a simpler form. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. A zero-divisor graph Λ (R) is an undirected graph on the vertex set R − { 0 } such that any two nonzero elements p , q ∈ R are adjacent whenever p ⋅ q = 0. The concept of entropy measure is fundamental and has been the subject of interest in various fields. The purpose of this paper is to calculate the entropy measure of Λ (R) for various rings. We take into consideration the ring ℤ h for h = φ ψ , φ 2 ψ , φ 2 ψ 2 , φ ψ χ , where φ , ψ and χ are distinct prime numbers. Moreover, we also discussed the entropy measure of the zero-divisor graph for the rings ℤ φ × ℤ ψ , ℤ φ × ℤ ψ × ℤ χ , ℤ φ ψ × ℤ χ , and ℤ φ 2 × ℤ ψ 2. This study will help us further understand the algebraic structure of the commutative rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, I an ideal of R , and M a finitely generated R -module. We consider the idealization R ⋉ M of M over R. The goal of this paper is to investigate algebraic properties of R ⋉ M that are related to those of R and M. Specifically, we provide characterizations for the Cohen–Macaulayness, sequentially Cohen–Macaulayness, generalized Cohen–Macaulayness, and graded maximal depth property of R ⋉ M with respect to I ⋉ M , in terms of the corresponding properties for R and M with respect to I. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, a proper ideal of R and N a nonzero finitely generated R -module with N ≠ N. Let d (respectively, c) be the smallest (respectively, greatest) non-negative integer i such that the local cohomology H i (N) is nonzero. In this paper, we provide sharp bounds under inclusion for the annihilators of the local cohomology modules H d (N) , H c (N) and these annihilators are computed in certain cases. Also, we construct a counterexample to Lynch's conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of ring called w p - S -Noetherian ring, which is a weak version of S -Noetherian ring property and study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamation of rings. Furthermore, we define the concept of nonnil w p - S -Noetherian ring property which is a generalization of the w p - S -Noetherian domain property and establish a characterization of this notion using pullbacks. [ABSTRACT FROM AUTHOR]

AbstractOur present work is motivated by several papers which have introduced and studied the notion of nonnil-Noetherian rings. Let A = ⊕α∈G Aα be a commutative ring with unity graded by an arbitrary commutative monoid G. We define A to be graded-nonnil-Noetherian ring if every nonnil graded ideal of A, that is, a graded ideal which is not contained in Nil(A), is finitely generated. The purpose of this paper is to give the graded version of several characterizations, properties and basic results proved in [3], [7] and [4]. We also show the non-triviality of our generalization by giving examples of graded-nonnil-Noetherian rings which are not nonnil-Noetherian and of graded-nonnil-Noetherian rings which are not graded-Noetherian. [ABSTRACT FROM AUTHOR]

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PRmodules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module M is PR-module if and only if for every g ∈ End(M), CM ∩ Tg is pure submodule of M⊕M (or CM ∩ Tg ≤PM⊕M for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules. [ABSTRACT FROM AUTHOR]

In this paper, using Gallai's Theorem and the notion of strong resolving graph, we determine the strong metric dimension in annihilating-ideal graph of commutative rings. For reduced rings, an explicit formula is given and for non-reduced rings, under some conditions, strong metric dimension is computed. [ABSTRACT FROM AUTHOR]

In this note we will show that finitely generated condition is necessary in Theorem 1.2 of the above mentioned paper. [ABSTRACT FROM AUTHOR]

Given a graph G whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling of G by the elements of R so that the difference of labels on adjacent vertices is an element of the corresponding edge ideal. The set of all generalized splines on a graph G with base ring R has a ring and an R -module structure. In this paper, we focus on the freeness of generalized spline modules over certain graphs with the base ring R = k [ x 1 , ... , x d ] where k is a field. We first show the freeness of generalized spline modules on graphs with no interior edges over k [ x , y ] such as cycles or a disjoint union of cycles with free edges. Later, we consider graphs that can be decomposed into disjoint cycles without changing the isomorphism class of the syzygy modules. Then we use this decomposition to show that generalized spline modules are free over k [ x , y ] and later we extend this result to the base ring R = k [ x 1 , ... , x d ] under some restrictions. [ABSTRACT FROM AUTHOR]

According to Dastanpour and Ghorbani, a ring R is said to satisfy divisibility on ascending chains of right ideals (ACCd) if, for every ascending chain of right ideals I1 ⊆ I2 ⊆ I3 ⊆ I4 ⊆ ... of R, there exists an integer k ∈ N such that for each i ≥ k, there exists an element ai ∈ R such that Ii = aiIi+1. In this paper, we examine the transfer of the ACCd-condition on ideals to trivial ring extensions. Moreover, we investigate the connection between the ACCd on ideals and other ascending chain conditions. For example we will prove that if R is a ring with ACCd on ideals, then R has ACC on prime ideals. [ABSTRACT FROM AUTHOR]

In this paper, author presents a generalization of an X-element in a multiplicative lattice L. For a particular M-closed subset X, author defines the concept of Φ -- r-element, Φ -- n-element, and Φ -- J-element. These elements generalize the notion of Φ -- r-ideals, Φ -- n-ideals, and Φ -- J-ideals of a commutative ring with unity to multiplicative lattices. An ideal I of a commutative ring R with unity is a Φ -- n-ideal (Φ -- Jideal) of R if and only if I is a Φ -- n-element (Φ -- J-element) of Id(R), the ideal lattice of R is proved. [ABSTRACT FROM AUTHOR]

For a connected graph G (V , E) a clique S ⊆ V (G) is twin-free if every pair of elements of S have distinct closed neighborhoods and the number of elements in a twin-free clique of maximum cardinality is called twin-free clique number of G. The annihilator graph A G (R) of a commutative and unital ring R is a graph whose vertices are all non-zero zero-divisors of R and there is an edge between two distinct vertices a , b if and only if ann (a) ∪ ann (b) is properly contained in ann (a b). In this paper, twin-free clique number of A G (R) is computed and as an application the strong metric dimension of A G (R) is characterized. Among other things, for a reduced ring R , the forcing strong metric dimension of A G (R) is given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend the domain of the Grace's theorem, Schur-Szegö composition theorem and Cohn-Egerváry theorem from the set of complex numbers to the set of bicomplex numbers. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the algebraic structure and theorems for constructing DNA codes of double cyclic codes of length (α, β) over the finite commutative chain ring 픽2+u픽2 with u2 = 0, where α and β are odd and even positive integers, respectively. Our main objectives include studying the structural properties of these codes, determining their generator polynomials and generating DNA codes from these codes. We investigate theorems for two types of double cyclic codes of length (α, β) over 픽2 + u픽2, ideal for generating DNA codes. These codes utilise non-separable structures and satisfy reverse and reverse-complement constraints. Additionally, we propose DNA codes derived from our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider flat epimorphisms of commutative rings R → U such that, for every ideal I ⊂ R for which I U = U , the quotient ring R / I is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the R-module U does not exceed 1. We also describe the Geigle–Lenzing perpendicular subcategory U ⊥ 0 , 1 in R -Mod. Assuming additionally that the ring U and all the rings R / I are perfect, we show that all flat R-modules are U-strongly flat. Thus, we obtain a generalization of some results of the paper [6], where the case of the localization U = S − 1 R of the ring R at a multiplicative subset S ⊂ R was considered. [ABSTRACT FROM AUTHOR]

A proper ideal I of a commutative ring R is said to be a strongly quasi-primary ideal if, whenever a , b ∈ R with a b ∈ I , then a 2 ∈ I or b ∈ I (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of R , called the class of quasi- n -ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of R is a maximal or a minimal ideal of R. Many examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {G}_R $ G R denote a unitary one-matching bi-Cayley graph over a finite commutative ring R. In this paper, we give a necessary and sufficient condition for $ \mathcal {G}_R $ G R (respectively, the complement of $ \mathcal {G}_R $ G R and the line graph of $ \mathcal {G}_R $ G R ) to be Ramanujan. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and Nil (R) be the ideal of all nilpotent elements of R. Let (R) = { I : I be a nontrivial ideal of R and there exists a nontrivial ideal J such that I J ⊆ Nil (R) }. The nil-graph of ideals of R is defined as the graph N (R) whose vertex set is the set (R) and two distinct vertices I and J are adjacent if and only if I J ⊆ Nil (R). A subset of vertices S ⊆ V (G) resolves a graph G , and S is a resolving set of G , if every vertex is uniquely determined by its vector of distances to the vertices of S. In particular, for an ordered subset S = { v 1 , v 2 , ... , v k } of vertices in a connected graph G and a vertex v ∈ V (G) \ S of G , the metric representation of v with respect to S is the k -vector D (v | S) = (d (v , v 1) , d (v , v 2) , ... , d (v , v k)). The set S is a resolving set for G if D (u | S) = D (v | S) implies that u = v , for all pair of vertices, v , u ∈ V (G) \ S. A resolving set S of minimum cardinality is the metric basis for G , and the number of elements in the resolving set of minimum cardinality is the metric dimension of G. If D (u | S) ≠ D (v | S) for every pair u , v of adjacent vertices of G , then S is called a local metric set of G. The minimum k for which G has a local metric k -set is the local metric dimension of G , which is denoted by lmd (G). In this paper, we determine metric dimension and local metric dimension of nil-graph of ideals of commutative rings. [ABSTRACT FROM AUTHOR]

Let (R, 픪) be a unitary commutative local Noetherian ring and I ⊂ R be an ideal. The aim of this paper is twofold: In the first part of this paper, we consider locally cohomologically complete intersection ideals of pure height dim R − 2. Next, we deal with cofinite modules. In particular, we investigate conditions for cofiniteness of HIh(R),h =ht(I) and conditions under which, the modules ExtRi(N,H픪dim R(R)) are of finite length for all i > 0 and all finitely generated R-module N. [ABSTRACT FROM AUTHOR]

One of the important classes of modules is the class of multiplication modules over a commutative ring. This topic has been considered by many authors and numerous results have been obtained in this area. After that, Tuganbaev also considered the multiplication module over a noncommutative ring. In this paper, we continue to consider the automorphism-invariance of multiplication modules over a noncommutative ring. We prove that if R is a right duo ring andM is a multiplication, finitely generated right R-module with a generating set {m1, . . ., mn} such that r(mi) = 0 and [miR: M] ⊆ C(R) the center of R, then M is projective. Moreover, if R is a right duo, left quasi-duo, CMI ring and M is a multiplication, non-singular, automorphism-invariant, finitely generated right R-module with a generating set {m1, . . ., mn} such that r(mi) = 0 and [miR: M] ⊆ C(R) the center of R, then MR ~= R is injective. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new property of rings called the f(x), g(x)-clean property, which is a generalization of the g(x)-clean property. We prove that a commutative ring Q is feebly clean if and only if it is f(x), f(x)-clean, where f(x) is a suitable quadratic polynomial. We prove additional results of this property and include examples for clarity. [ABSTRACT FROM AUTHOR]

For a commutative ring R with unity, the cozero-divisor graph denoted by Γ'(R), is an undirected simple graph whose vertex set is the set of all non-zero and non-unit elements of R. Two distinct vertices x and y are adjacent if and only if x does not belong to the ideal Ry and y does not belong to Rx. The cozero-divisor graph on the ring of integers modulo n is a generalized join of its induced sub graphs all of which are null graphs. This property of the cozero-divisor graph on Zn is used in finding its distance spectrum. In this paper, the distance matrix of the cozero-divisor graph on the ring of integers modulo n is discovered and the general method is discussed to find its distance spectrum, for any value of n. Also, the distance spectrum of this graph is explored for some values of n, by means of the vertex weighted distance matrix of the co-proper divisor graph of n. [ABSTRACT FROM AUTHOR]

Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

Over the years, prime submodules and their generalizations have played a pivotal role in commutative algebra, garnering considerable attention from numerous researchers and scholars in the field. This papers presents a generalization of 1-absorbing primary ideals, namely the classical 1-absorbing primary submodules. Let ℜ be a commutative ring and M an ℜ-module. A proper submodule K of M is called a classical 1-absorbing primary submodule of M, if x y z η ∈ K for some η ∈ M and nonunits x , y , z ∈ ℜ , then x y η ∈ K or z t η ∈ K for some t ≥ 1 . In addition to providing various characterizations of classical 1-absorbing primary submodules, we examine relationships between classical 1-absorbing primary submodules and 1-absorbing primary submodules. We also explore the properties of classical 1-absorbing primary submodules under homomorphism in factor modules, the localization modules and Cartesian product of modules. Finally, we investigate this class of submodules in amalgamated duplication of modules. [ABSTRACT FROM AUTHOR]

Prime ideals and their generalizations are crucial in numerous research areas, particularly in commutative algebra. The concept of generalization of prime ideals begins with the study of weakly prime ideals. Since then, subsequent works aimed at expanding this concept into more generalized forms. Among these, S-prime ideals and 2-prime ideals have reaped attention recently. This paper aims to characterize S-2-prime ideals, which serve as a generalization encompassing both 2-prime ideals and S-prime ideals. To accomplish this objective, we construct an ideal which distinct from a multiplicatively closed subset with the help of commutative rings. We investigate the localization and the S-2-prime avoidance lemma in commutative rings. Furthermore, we explore the properties of this class of ideals in trivial ring extensions and amalgamated algebras along an ideal. We delve into S-properties for compactly packedness, compactly 2-packedness and coprimely packedness in trivial ring extentions. Moreover, this notion of ideals helps us to indicate that many results stated in S-prime ideals and 2-prime ideals can be readily expanded to the framework of S-2-prime ideals. Supporting examples also highlight a significant distinction between S-2-prime ideals and stated ideals. [ABSTRACT FROM AUTHOR]

The call for papers welcomed any original paper within the scope and not simultaneously submitted to another journal or conference. Cristina Bertone, Universitá di Torino, Italy Michela Ceria, Politecnico di Bari, Italy Fatma Karaoglu, Gebze Technical University, Turkey Ilias Kotsireas, Wilfrid Laurier University, Canada Teo Mora, University of Genova, Italy Dimitris Simos, SBA Research Center, Austria Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This volume contains the refereed papers related to the 26th International Conference on Applications of Computer Algebra, which has been held online, from the 23rd to the 27th of July 2021. [Extracted from the article]

An abelian group A is called a T I -group if every associative ring with the additive group A is filial. The filiality of a ring R means that the ring R is associative and all ideals of any ideal of R are ideals in R. In this paper, torsion-free T I -groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition T I implies the indecomposability and homogeneity. The paper contains constructions of 2 ℵ 0 such groups of any rank from 2 to 2 ℵ 0 which are pairwise non-isomorphic. [ABSTRACT FROM AUTHOR]

In this article, we present the generalized ρ dependent polynomials for the calculations of eccentricity, distance, total distance, and degree-based topological indices of the identity graph of Zρ. This is a thorough work in which we present many topological indices and coindices as ρ dependent polynomials. The polynomials presented here can play a key role in the further development of the theory of topological indies for commutative rings. This paper presents a brand new approach to generalizing the topological indices because instead of the traditional way. A set-theoretic method is introduced here that can be very helpful and game-changing in the field of algebraic graph theory first of all sets of vertices for the identity graph of the commutative ring Zρ are partitioned into various sets, which makes it easier to generalize the degrees, distances, and eccentricities of this graph. This paper presents various results that make it easier to generalize the topological indices of the identity graph of Zρ. [ABSTRACT FROM AUTHOR]

Let I be the commutative non-unital ring of order 4 defined by generators and relations. I = a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0. Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths n = 6 , and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths n = 7 and 8, and describe five new variants of the two building-up construction methods. We find that when n = 8 there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions. [ABSTRACT FROM AUTHOR]

Let Γ be a group, A be a Γ-graded commutative ring with unity 1, and D a graded A-module. In this paper, we introduce the concept of graded weakly Jgr-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly Jgr-semiprime submodules. For example, we give a characterization of graded weakly Jgr-semiprime submodules. Also, we find some relations between graded weakly Jgr-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly Jgr-semiprime submodules are investigated. A proper graded submodule U of D is said to be a graded weakly Jgr-semiprime submodule of D if whenever rg ∈ h(A); mh ∈ h(D) and n ∈ Z+ with 0 ≠ rngmh ∈ U, then rgmh ∈ U + Jgr(D), where Jgr(D) is the graded Jacobson radical of D. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with zero-divisors Z (R) and i be a positive integer. The i -extended zero-divisor graph of R , denoted by Γ ¯ i (R) , is the (simple) graph with vertex set Z (R) ∗ = Z (R) \ { 0 } , the set of nonzero zero-divisors of R , and two distinct nonzero zero-divisors x and y are adjacent whenever there exist two positive integers n , m ≤ i such that x n y m = 0 with x n ≠ 0 and y m ≠ 0. The i -extended zero-divisor graph of R is well studied in 10. In this paper, we characterize the i -extended zero-divisor graphs of idealizations R ⋉ M (where M is an R -module). Namely, we study in detail the behavior of the filtration ( Γ ¯ i (R ⋉ M)) i ∈ ℕ ∗ as well as the relations between its terms. We also characterize the girth and the diameter of Γ ¯ i (R ⋉ M) and we give answers to several interesting and natural questions that arise in this context. [ABSTRACT FROM AUTHOR]

The comaximal graph Γ (R) of a commutative ring R is a simple graph with vertex set R and two distinct vertices u and v of Γ (R) are adjacent if and only if u R + v R = R. In this paper, we find the sharp bounds for the Sombor index for comaximal graphs of integer modulo ring ℤ n and give the corresponding extremal graphs. Also, we find the Sombor eigenvalues and the bounds for the Sombor energy of comaximal graphs of ℤ n . [ABSTRACT FROM AUTHOR]

This paper deals with some results concerning finitely generated coreduced comultiplication modules over a commutative ring. [ABSTRACT FROM AUTHOR]

Let A = ⊕ α ∈ Γ A α and B = ⊕ α ∈ Γ B α be two commutative rings graded by an arbitrary commutative monoid Γ , let J be a homogeneous ideal of B and f : A → B a graded ring homomorphism. In this paper, we introduce a new graded ring construction called "the graded amalgamation of A with B along J with respect to f " (denoted by A ⋈ f J). Then we investigate, through the properties of the homogeneous components that build the body of this "graded" study, the transfer of the property of graded-coherence to the graded amalgamation A ⋈ f J. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring and α be an automorphism of R. A mapping F : R → R (not necessarily additive) is called multiplicative generalized (α, α)- derivation if there exists a unique (α, α)-derivation d of R such that F(xy) = F(x)α(y) + α(x)d(y) for all x, y = R. In the present paper, we intend to study several algebraic identities involving multiplicative generalized (α, α)-derivations on appropriate subsets of semiprime rings and collect the information about the commutative structure of these rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and Z (R) be its zero-divisors set. The zero-divisor graph of R , denoted by Γ (R) , is an undirected graph with vertex set Z (R) ∗ = Z (R) ∖ { 0 } and two distinct vertices a and b are adjacent if and only if a b = 0. In this paper, for n = p R q S where p and q are primes (p < q) and R and S are positive integers, we calculate the A α spectrum of the graphs Γ (ℤ n). [ABSTRACT FROM AUTHOR]

This paper studies λ-constacyclic codes and skew λ-constacyclic codes over the finite commutative non-chain ring R = Z4 + uZ4 + u²Z4 with u³ = 0 for λ = (1 + 2u + 2u²) and (3 + 2u + 2u²). We introduce distinct Gray maps and show that the Gray images of λ-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over Z4. It is also shown that the Gray images of skew λ-constacyclic codes are quasi-cyclic codes of length 2n and index 2 over Z4. Moreover, the structure of λ-constacyclic codes of odd length n over the ring R is determined and give some suitable examples. [ABSTRACT FROM AUTHOR]