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Let 픞 be an ideal of a commutative noetherian ring R and M,N two R-modules with M finitely generated. It is shown that if either H픞i(N) is an 픞-cofinite module of dimension ≤ 1 for all i, or 픞 is a principal ideal and ExtRi(R/픞,N) is finitely generated for all i, or ExtRi(R/픞,N) is finitely generated and dimRH픞i(M,N) ≤ 1 for all i, then the R-module H픞t(M,N) is 픞-cofinite for all t ≥ 0. [ABSTRACT FROM AUTHOR]

In this paper we study the Diophantine problem in Chevalley groups G π (Φ , R) , where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1. We establish a variant of double centralizer theorem for elementary unipotents x α (1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup X α , α ∈ Φ , is Diophantine in G. Then we prove that the Diophantine problem in G π (Φ , R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. In this paper, we find out close relationships between the module M being embedded in a module of projective dimension at most n and the (n + 1) -torsionfreeness of the n th syzygy of M. As an application, when R is local with residue field k , we compute the dimensions as k -vector spaces of Ext modules related to syzygies of k. [ABSTRACT FROM AUTHOR]

Let p be a prime such that pm ≡ 1(mod4), where m is a positive integer. For any nonzero element α of 픽pm, we determine the algebraic structure of all α-constacyclic codes of length 4ps over the finite commutative chain ring 픽pm[u] 〈u3〉 ≅픽pm + u픽pm + u2픽 pm, where u3 = 0 and s is a positive integer. If the unit α ∈ 픽pm is a square, α = δ2, each α-constacyclic code of length 4ps is expressed as a direct sum of an − δ-constacyclic code and an δ- constacyclic code of length 2ps. In the main case that the unit α is not a square, it is shown that any nonzero polynomial of degree at most 3 over 픽pm is invertible in the ambient ring (픽pm+u픽pm+u2픽pm)[x] 〈x4ps−α〉. It is also proven that the ambient ring (픽pm+u픽pm+u2픽pm)[x] 〈x4ps−α〉 is a local ring with the unique maximal ideal 〈x4 − α 0,u〉, where α0ps = α. Such α-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each α-constacyclic code are obtained. The non-existence of self-dual and isodual α-constacyclic codes of length 4ps over 픽pm + u픽pm + u2픽 pm, when the unit α is not a square, is likewise proved. [ABSTRACT FROM AUTHOR]

Let G be a group with identity and R be a G-graded commutative ring with nonzero identity. A proper graded ideal I of R is called a graded 1-absorbing prime ideal (respectively, graded 1-absorbing primary ideal) if whenever nonunit homogeneous elements a,b,c ∈ R with abc ∈ I, then ab ∈ I or c ∈ I (respectively, ab ∈ I or c ∈ Gr(I), where Gr(I) is the graded radical of I). The purpose of this paper is to study the transfer of some graded 1-absorbing-like properties to the graded amalgamated algebra along an ideal (denoted by A⋈fJ). Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and I be a proper ideal of a commutative ring R. The ideal-based triple zero-divisor graph of a commutative ring is a graph, denoted by TZΓI(R), with the vertex set TZI(R) = {a ∈ R : there existsb,c ∈ Rsuch thatabc ∈ I,ab∉I,bc∉I,ac∉I} and two vertices a,b are adjacent if and only if there is a c such that abc ∈ I,ab∉I,bc∉I,ac∉I. In this paper, we discuss the connectedness, diameter, girth of TZΓI(R). We classify all finite commutative rings for which TZ(ΓI(R)) is either complete, unicyclic or split graph. Also, we characterize all finite commutative rings for which TZΓI(R) is perfect. Finally, we classify all finite commutative rings for which TZΓI(R) has genus at most one. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new concept called the tripotent divisor graph of a commutative ring. It is defined with vertices set in a ring R, where distinct vertices r1 and r2 are connected by an edge if their product belongs to the set of all nonunite tripotent in R. We denote this graph as 3I Γ(R). We utilize this graph to examine the role of tripotent elements in the structure of rings. Additionally, we provide various findings regarding graph‐theoretic characteristics of this graph, including its diameter, vertex degrees, and girth. Furthermore, we investigate the size, central vertices, and distances between vertices for the tripotent divisor graph formed by the direct product of two fields. [ABSTRACT FROM AUTHOR]

The notion of the metric dimension of a graph is well-known and its study is well entrenched in the literature. In this paper, we introduce new classes of graphs that exhibit remarkable characteristic: the metric dimension and the diameter of the unit graph are equal. Additionally, we provide a characterization of finite commutative rings R, wherein the metric dimension assumes a value k, where 1 ≤ k ≤ 3. Further, we also demonstrate an exhaustive examination that ascertains the precise finite commutative rings R, in which domination number is equal to metric dimension of unit graph. [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]

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]

We present a proof that all straight domains are locally divided—thereby answering two open problems posed by Dobbs and Picavet, which appeared in the survey "Open Problems in Commutative Ring Theory" written by Cahen, Fontana, Frisch, and Glaz. In fact, we are able to prove a stronger result: a prime ideal of a domain is straight if and only if it is locally divided. [ABSTRACT FROM AUTHOR]

In this article, we show that Mori domains, pseudo-valuation domains, and n-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain R is a Mori locally pseudo-valuation domain if and only if each proper ideal of R is a finite product of 2-absorbing ideals of R. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form A + XB [ X ] where A is a subring of a commutative ring B and X is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of n-absorbing ideals for some n ≥ 2 if and only if A ⊆ B is essentially a finite product of field extensions. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given. [ABSTRACT FROM AUTHOR]

Let R be an associative and 2-torsion-free ring with an identity. in this work, we will generaliz the results of differentially prime rings in [18] by applying the hypotheses in a differentially semiprime rings. In particular, we have proved that if R is a D-semiprime ring, then either R is a commutative ring or D is a semiprime ring. [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]

A total coloring of a graph G is an assignment of colors to all the elements (vertices and edges) of the graph in such a way that no two adjacent or incident elements receive the same color. The total chromatic number of G, denoted by χzz(G), is the minimum number of colors needed for a total coloring of G. The Total Coloring Conjecture (TCC) proposed independently by Behzad and Vizing claims that, Δ(G) + 1 ≤ χ"(G) ≤ Δ(G) + 2, where Δ(G) is the maximum degree of G. The lower bound is sharp and the upper bound remains to be proved. In this paper, we prove the TCC for certain classes of lexicographic and deleted lexicographic products of graphs. Also, we obtained the lower bound for certain classes of these products. [ABSTRACT FROM AUTHOR]

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 = 1 ") we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull's lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.1 See https://github.com/iblech/constructive-maximal-ideals/. This text is a revised and extended version of the conference paper (In Revolutions and Revelations in Computability. 18th Conference on Computability in Europe (2022) Springer). The conference paper only briefly sketched the connection with dynamical algebra; did not compare this connection with other flavors of set-theoretic forcing; and sticked to the case of commutative algebra only, passing on the generalization to inconsistency predicates and well-orders. [ABSTRACT FROM AUTHOR]

Let K be a proper ideal of a commutative ring S. Then K is a strongly irreducible (SI) ideal if for any two ideals A and B of S, ANB⊆ K implies A Kor BK. We say a ring is strongly irreducible (SI) if all its proper ideals are strongly irreducible. In this paper, some properties and characterizations of such rings are given. The relations between SI rings and some types of rings are also studied. For an SI ring S, the S strongly irreducible spectrum X = S.spec(s) of S is the set S. spec(S)= {I: I is an ideal of S} and the S variety of a subset E of S is the set V (E) = {JE S. spec(S): E1}. Then the family F = {V(E): ES} satisfies the axioms for closed sets of a topology on X = S.spec(S). Consequently, if X(E) = S. spec(S)\V(E), then the family H = {X(E): E S} forms a topology on X = S. spec(S). This topology is said to be S. spec(S) topology or S Zariski topology. In this work, some properties of S. spec(S) topology are also studied. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and S a multiplicatively closed subset of R. Hamed and Malek [7] defined an ideal P of R disjoint with S to be an S-prime ideal of R if there exists an s ∈ S such that for all a, b ∈ R if ab ∈ P, then sa ∈ P or sb ∈ P. In this paper, we introduce the notions of S-k-prime and S-k-semiprime ideals of semirings, S-k-m-system, and S-k-p-system. We study some properties and characterizations for S-k-prime and S-k-semiprime ideals of semirings in terms of S-k-m-system and S-k-p-system respectively. We also introduce the concepts of S-prime semiring and S-semiprime semiring and study the characterizations for S-k-prime and S-k-semiprime ideals in these two semirings. [ABSTRACT FROM AUTHOR]

The mz-elements of a coherent quantale have recently been defined by the author as an abstraction of the mz-ideals of a unital commutative ring. Having as its starting point the Dube and Ighedo recent paper on higher order ideals in ring theory, this paper deals with the higher order mz-elements of a coherent quantale A. For each natural number n we define the mzn-elements of A, so we obtain an ascending sequence that covers the set of all higher order mz-elements. We obtain a lot of properties of this sequence. In particular, the stationarity of the sequence is studied. Another category of results investigates how the coherent quantale morphisms preserve such properties. [ABSTRACT FROM AUTHOR]

Let G be a group, R be a G-graded commutative ring with identity, M be a unitary graded R-module, Specg(R) be the set of graded prime ideals of R, and Cl.Specg(M) be the set of all graded classical prime submodules of M. In this paper among other things, the author studied the Zariski topology on both and Cl.Specg(M), and investigate some properties of the Zariski topology on Cl.Specg(M) and some conditions under which the graded classical prime spectrum of M is a spectral for its Zariski topology. [ABSTRACT FROM AUTHOR]

Graph theory has become a hot topic in Mathematics due to the gradual research done in graph theory. Product of graphs enables the combination or decomposition of its elemental structures. In graph theory there are four standard products, each with its own set of applications and theoretical interpretations. In this article, we study these graph products of zero-divisor graphs of commutative rings and determine their structural properties such as connectivity, diameter and girth. We also determine when the graph product of zero-divisor graphs of ring Zn and Zm are Eulerian. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following question: if all homology groups of a space X are finitely generated, and if R is a commutative ring with identity, is it true that the homology and cohomology R-modules Hi(X; R) and Hi(X; R) are also finitely generated? We show that the answer to this question is negative in general, but affirmative if R is an integral domain. In the case when R is a principal ideal domain, and Hi(X; R) is finitely generated for all i, we also discuss computing Hi(X; M) and Hi(X; M) for a finitely generated R-module M. [ABSTRACT FROM AUTHOR]

In this paper, prime ideal bundles and semi-prime and irreducible ideal bundles of a Lie algebra bundle are defined and their relation with prime ideal bundles is studied. [ABSTRACT FROM AUTHOR]

In this paper, we introduced m-extension dual complex Fibonacci p-numbers. We established the properties of mextension dual complex Fibonacci p-numbers. They are connected to complex Fibonacci numbers, complex Fibonacci p-numbers and dual complex Fibonacci p-numbers. [ABSTRACT FROM AUTHOR]

Let R be a commutative semiring and I be a k-ideal of R. In this paper, we introduce the k-ideal-based graph of R, denoted by ΓI ∗ (R). The basic properties and possible structures of the graph are studied. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I, J be two ideals of R, and M, N be two R-modules. Let S be a Serre subcategory of the category of R-modules. We introduce Serre cohomological dimension of N, M with respect to (I, J), as cdS(I, J, N, M) = sup{i ∈ N0 : HiI,J (N, M) 6∈ S}. We study some properties of cdS(I, J, N, M), and we get some formulas and upper bounds for it. [ABSTRACT FROM AUTHOR]

Recently, quasisymmetric functions have been widely studied due to their big connection to enumerative combinatorics, combinatorial Hopf algebra and number theory. The Bayer filter mosaic, named due to Bryce Bayer (1929-2012), is a color filter array used to arrange RGB color filters on a square grid of photosensors. It is the most common pattern of filters, and almost all professional digital cameras are applications of this filter. We use this filter to introduce the Bayer Noise quasisymmetric functions, and we study some combinatorial algebraic and coalgebraic structures on Quasi-Bayer Noise modules and on Quasi-Bayer GB-Noise modules. We explicitly describe the primitive basis elements for each comultiplication defined on Quasi-Bayer Noise modules, and we calculate different kinds of comultiplications defined on Quasi-Bayer Noises module over a fixed commutative ring k. [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]

Let A,B be two rings and n ⩾ 2 be an integer. An additive map h: A → B is called an n-Jordan homomorphism if h(xn) = h(x)n for all x ∈ A; h is called an n-homomorphism or an anti-n-homomorphism if h(Πni=1 xi) = Πn i=1 h(xi) or h(Πni=1 xi) = Πn i=0 h(xn-i ∈ A. We give the following variation of a theorem on n-Jordan homomorphisms due to I.N. Herstein: Let n ≥ 2 be an integer and h be an n-Jordan homomorphism from a ring A into a ring B of characteristic greater than n. Suppose further that A has a unit e, then h = h(e)τ, where h(e) is in the centralizer of h(A) and τ is a Jordan homomorphism. By using this variation, we deduce the following result of G. An: Let A and B be two rings, where A has a unit and B is of characteristic greater than an integer n ≥ 2. If every Jordan homomorphism from A into B is a homomorphism (anti-homomorphism), then every n-Jordan homomorphism from A into B is an n-homomorphism (anti-n-homomorphism). As a consequence of an appropriate lemma, we also obtain the following result of E. Gselmann: Let A,B be two commutative rings and B is of characteristic greater than an integer n ≥ 2. Then every n-Jordan homomorphism from A into B is an n-homomorphism. [ABSTRACT FROM AUTHOR]1, ..., xn ∈ A. We give the following variation of a theorem on n-Jordan homomorphisms due to I.N. Herstein: Let n ≥ 2 be an integer and h be an n-Jordan homomorphism from a ring A into a ring B of characteristic greater than n. Suppose further that A has a unit e, then h = h(e)τ, where h(e) is in the centralizer of h(A) and τ is a Jordan homomorphism. By using this variation, we deduce the following result of G. An: Let A and B be two rings, where A has a unit and B is of characteristic greater than an integer n ≥ 2. If every Jordan homomorphism from A into B is a homomorphism (anti-homomorphism), then every n-Jordan homomorphism from A into B is an n-homomorphism (anti-n-homomorphism). As a consequence of an appropriate lemma, we also obtain the following result of E. Gselmann: Let A,B be two commutative rings and B is of characteristic greater than an integer n ≥ 2. Then every n-Jordan homomorphism from A into B is an n-homomorphism. [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]

We present a quick introduction to Frobenius complexity and gradings with rational twist for commutative rings of prime characteristic, providing a place to start for those interested in exploring these topics. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of multiplicative n-th root functions n√ over finite semigroups, in order to implement these ideas on finite groups, fields and commutative rings. A set of sufficient and necessary conditions are presented for existence of multiplicative n-th root functions over different algebraic structures. It is also shown that once the existence is established, the uniqueness is guaranteed. In addition, we describe the construction procedure of such a function. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and M an R -module. We say that M satisfies strong accr ∗ if for every submodule N of M and for every sequence 〈 r n 〉 of elements of R , the ascending sequence of submodules of the form, N : M r 1 ⊆ N : M r 1 r 2 ⊆ N : M r 1 r 2 r 3 ⊆ ⋯ is stationary. We say that a ring R satisfies strong accr ∗ if R regarded as a module over R satisfies strong accr ∗. In this paper, we give a necessary and sufficient condition for the pulback (respectively, the Nagata's idealization R (+) M) to be strong accr ∗ ring. We also prove a new characterization for a valuation ring to be strong accr ∗. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. For a positive integer [Formula: see text] let [Formula: see text] be the set of all n × n matrices over R and [Formula: see text] be the set of all non-zero matrices of [Formula: see text] The zero-divisor graph of [Formula: see text] is a simple directed graph with vertex set the non-zero zero-divisors in [Formula: see text] and two distinct matrices A and B are adjacent if their product is zero. Given a matrix [Formula: see text] Tr(A) is the trace of the matrix A. The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] [Formula: see text] and two distinct vertices A and B are adjacent if and only if Tr [Formula: see text] For an ideal I of R, the notion of the ideal based trace graph, denoted by [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and two distinct vertices A and B are adjacent if and only if Tr [Formula: see text] In this survey, we present several results concerning the zero-divisor graph, trace graph and the ideal based trace graph of matrices over R.

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]

Let $({\cal{A}},{\cal{E}})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of ${\operatorname{Ext}}_{\cal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and find a range of applications from detecting regularity to understanding Ulrich modules. [ABSTRACT FROM AUTHOR]

Let G be an abelian group with identity 0 and let R be a commutative graded ring of type G with nonzero unity. Let I(R) be the set of all ideals of R and let δ : I(R) −→ I(R) be a function. Then, according to (R. Abu-Dawwas, M. Refai, Graded δ-Primary Structures, Bol. Soc. Paran. Mat., 40 (2022), 1-11), δ is called a graded ideal expansion of a graded ring R if it assigns to every graded ideal I of R another graded ideal δ(I) of R with I ⊆ δ(I), and if whenever I and J are graded ideals of R with J ⊆ I, we have δ(J) ⊆ δ(I). Let δ be a graded ideal expansion of a graded ring R. In this paper, we introduce and investigate a new class of graded ideals that is closely related to the class of graded δ-primary ideals. A proper graded ideal I of R is said to be a graded 1-absorbing δ-primary ideal if whenever nonunit homogeneous elements a, b, c ∈ R with abc ∈ I, then ab ∈ I or c ∈ δ(I). After giving some basic properties of this new class of graded ideals, we generalize a number of results about 1-absorbing δ-primary ideals into these new graded structure. Finally, we study the graded 1-absorbing δ-primary ideals of the localization of graded rings and of the trivial graded ring extensions. [ABSTRACT FROM AUTHOR]

In this paper, we study graded pseudo 2-prime ideals of graded commutative rings with nonzero identities. Let G be a commutative additive monoid with an identity element 0, and R = ⊕ g∈G Rg be a commutative graded ring with a nonzero identity element. A proper graded ideal I of R is said to be a graded pseudo 2-prime ideal if whenever ab ∈ I for some homogeneous elements a, b ∈ R, then a2n ∈ In or b2n ∈ In for some n ∈ N. Besides giving many properties of graded pseudo 2-prime ideals, we characterize graded almost valuation domains in terms of our new concept. [ABSTRACT FROM AUTHOR]

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph E A G (R) associated to a commutative ring R , denoted by dim M (E A G (R)) , dim + (E A G (R)) and res (E A G (R)) , respectively. Here we prove the finiteness conditions of dim M (E A G (R)) and dim + (E A G (R)). In addition, we characterize dim M (E A G (R)) , dim + (E A G (R)) and res (E A G (R)) for artinian rings and the direct product of rings. [ABSTRACT FROM AUTHOR]

We study the structure of rings which satisfy the von Neumann regularity of commutators, and call a ring R C-regular if a b − b a ∈ (a b − b a) R (a b − b a) for all a , b in R. For a C-regular ring R , we prove J (R [ X ]) = N ∗ (R [ X ]) = N ∗ (R) [ X ] = W (R) [ X ] ⊆ Z (R [ X ]) , where J (A) , N ∗ (A) , W (A) , Z (A) are the Jacobson radical, upper nilradical, Wedderburn radical, and center of a given ring A , respectively, and A [ X ] denotes the polynomial ring with a set X of commuting indeterminates over A ; we also prove that R is semiprime if and only if the right (left) singular ideal of R is zero. We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular, from any given ring. Moreover, for a C-regular ring R , the following are proved to be equivalent: (i) R is Abelian; (ii) every prime factor ring of R is a duo domain; (iii) R is quasi-duo; and (iv) R / W (R) is reduced. [ABSTRACT FROM AUTHOR]

We introduce the notion of PC cancellative additive monoids with infinity and use it to characterize cancellative additive principal ideal domains with infinity. Our characterization improves various known characterizations from the literature, both, in the context of the commutative cancellative monoids, as well as in the context of the analogues of the statements from the commutative ring theory. [ABSTRACT FROM AUTHOR]

We show that any n-absorbing ideal must be strongly n-absorbing, which is the first of Anderson and Badawi's three interconnected conjectures on absorbing ideals. We prove this by introducing and studying objects called maximal and semimaximal collections, which are tools for analyzing the multiplicative ideal structure of commutative rings. Furthermore, we discuss and make heavy use of previous work on absorbing ideals done by Anderson and Badawi, Choi and Walker, Laradji, and Donadze, among others. At the end of this paper, we pose three conjectures, each of which implies the last unsolved conjecture made by Anderson and Badawi. [ABSTRACT FROM AUTHOR]

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel'd H-module algebras and the Brauer group BQ (R , H *) of Hopf Yetter-Drinfel'd H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S, H)-module algebras and BQ (S o p , H *) , the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S o p , H *) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel'd H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and M be an R-module. The R-module M is called distributive if for every submodules S, T and U of M, the equality S ∩ (T + U) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for M to be distributive based on injective envelopes. The proof uses Matlis' results on injective modules. [ABSTRACT FROM AUTHOR]

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems A 2 , F 4 , B l , C l ; with 1/2 and 1/3 for the system G 2 ) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However, the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description. [ABSTRACT FROM AUTHOR]