Search

Studying algebraic structures via graphs and hypergraphs assigned to them can be of interest. Especially, computing the genus of a graph as a topological index leads to a better understanding of the related algebraic structure. In this direction we apply a hypergraph, namely 3 -zero divisor hypergraph assigned to a commutative ring and study its genus. In this paper, we characterize all finite commutative nonlocal rings A with identity whose ℋ 3 (A) has genus two. Further, we classify all finite commutative nonlocal rings A whose ℋ 3 (A) has crosscap two. Moreover, we provide a MATLAB code for calculating 3 -zero-divisor of ℤ n and the hyperedge of 3 -zero-divisor hypergraph of  ℤ n . [ABSTRACT FROM AUTHOR]

Let Γ = (V , E) be a finite undirected graph without loops or multiple edges. A non-empty set of vertices S ⊆ V is called powerful alliance if for every vertex u ∈ N [ S ] , | N [ u ] ∩ S | ≥ | N [ u ] ∩ S ¯ |. A powerful alliance dominating set is called global. The global powerful alliance number γ a p (Γ) is defined as the minimum cardinality among all global powerful alliances. In this paper, we initiate the study of the global powerful alliance number of zero-divisor graphs Γ (R) with R is a finite commutative ring. Hence, we calculate γ a p (Γ (R)) for some usual kind of finite rings. As application, we give the global powerful alliance number of all zero-divisor graphs of finite commutative rings of order ≤ 7. [ABSTRACT FROM AUTHOR]

In this paper, we call a right module M (strongly) virtually regular if every (finitely generated) cyclic submodule of M is isomorphic to a direct summand of M. M is said to be completely virtually regular if every submodule of M is virtually regular. In this paper, characterizations and some closure properties of the aforementioned modules are given. Several structure results are obtained over commutative rings. In particular, the structures of finitely presented (strongly) virtually regular modules and completely virtually regular modules are fully determined over valuation domains. Namely, for a valuation domain R with the unique nonzero maximal ideal P, we show that finitely presented (strongly) virtually regular modules are free if and only if P is not principal; and that P = Rp is principal if and only if finitely presented virtually regular modules are of the form Rn ⊕R Rpn1 ⊕ R Rp2n2 ⊕⋯ ⊕ R Rpknk for nonnegative integers n,k,n1,n2,…,nk. Similarly, we prove that P = Rp is principal if and only if finitely presented strongly virtually regular modules are of the form Rn ⊕ ( R Rp)m, where m,n are nonnegative integers. We also obtain that, R admits a nonzero finitely presented completely virtually regular module M if and only if P = Rp is principal. Moreover, for a finitely presented R-module M, we prove that: (i) if R is not a DVR, then M is completely virtually regular if and only if M≅( R Rp)m; and (ii) if R is a DVR, then M is completely virtually regular if and only if M≅Rn ⊕ ( R Rp)m. Finally, we obtain a characterization of finitely generated virtually regular modules over the ring of integers. [ABSTRACT FROM AUTHOR]

In this paper, we continue to study weakly uniserial modules and rings. A module is called weakly uniserial if its submodules are comparable regarding embedding. Also, a right weakly uniserial ring is a ring which is weakly uniserial as a right module over itself. In addition to providing more properties of these modules, we introduce and investigate weakly serial modules (which are a direct sum of weakly uniserial modules). A right weakly serial ring is a ring which is weakly serial as a right module over itself. It is shown that over a right Artinian local ring R, every (2-generated) right R-module with a semiprime annihilator is weakly uniserial. The converse is true when R is a commutative Noetherian ring. We prove that an injective ℤ-module is weakly uniserial if and only if it is isomorphic to ℤp∞, for some prime number p. In a weakly serial module we show that every its nonzero submodule contains a weakly uniserial submodule and every its fully invariant submodule is weakly serial. Also, it is shown that a right duo right self-injective ring is right weakly serial if and only if every projective right module is (weakly) serial. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and S be a multiplicative subset of R. In this paper, we introduce and study an S-version of valuation rings, that is called the notion of S-valuation rings. Therefore, we say that R is an S-valuation ring if for all (a,b) ∈ R2, there exists an element s ∈ S such that saR ⊆ bR or sbR ⊆ aR. Moreover, we study the transfer of this property in trivial ring extensions and amalgamated algebras along an ideal. Finally, we explore the graded version of this concept in rings graded by a torsionless grading monoid. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate two weak versions of graded-Noetherian modules. Let G be an abelian group with an identity element denoted by 0, R be a commutative G-graded ring and M be a G-graded R-module. We say that M is a graded-r-Noetherian module if every graded-r-submodule of M is finitely generated. Also, we say that M is a graded-weakly-Noetherian module if all finitely generated graded submodules of M are graded-Noetherian R-modules. We give many properties of the two different concepts and we examine the relation between them and the different concepts that already exist in the literature. We illustrate our study by giving many nontrivial examples and counter-examples. Moreover, we characterize graded-Noetherian modules in terms of graded-r-Noetherian and graded-weakly-Noetherian modules. Finally, we examine the transfer of these two concepts in the graded idealization of graded modules. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring of dimension d. First, we define the “geometric subring” A of a polynomial ring R[T] or R[T,T−1] of dimension d + 1 (the definition of geometric subring is more general, see (1.2)). Then we prove that every locally complete intersection ideal of height d + 1 is a complete intersection ideal. Thus, improving the general bound of Mohan Kumar [On two conjectures about polynomial rings, Invent. Math. 46(3) (1978) 225–236] for an arbitrary ring of dimension d + 1. Afterward, we deduce that every finitely generated projective A-module of rank d + 1 splits off a free summand of rank one. This improves the general bound of Serre [J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle. Séminaire Dubreil. Algébre et théorie des nombres, Volume 11 (1957–1958) no. 2, Talk No. 23, p. 18] for an arbitrary ring. Finally, applications are given to a set-theoretic generation of an ideal in the geometric ring A and its polynomial extension A[X]. [ABSTRACT FROM AUTHOR]

An error in Corollary 9.32 of [C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, Vol. 65 (American Mathematical Society, Providence, RI, 2004)], motivated us to consider again FPF rings which were initiated by Faith in the 1970s. In this paper, it is shown that a commutative ring R is reduced FPF if and only if it is Π -semihereditary. We show that when a semiperfect ring with a strongly right bounded basic ring with right and left Ore conditions, is an FPF ring. After some general results, the article focuses on rings of continuous functions. We give some algebraic characterizations for a C (X) to be FPF and retrieve a result of Jorge Martinez. Also, we show that a space X is fraction-dense if and only if Q cl (X) is a continuous ring. [ABSTRACT FROM AUTHOR]

The prime ideal sum graph of a commutative unital ring R , denoted by PIS (R) , is an undirect and simple graph whose vertices are nontrivial ideals of R and there exists an edge between two distinct vertices if and only if their sum is a prime ideal of R. In this paper, the metric dimension of PIS (R) is discussed and some formulae for this parameter in PIS (R) are given. [ABSTRACT FROM AUTHOR]

We explore the concept of the S-Krull dimension in commutative rings, extending classical notions of Krull dimension by incorporating multiplicative subsets S of a ring R. We provide characterizations of S-prime, S-maximal, and S-minimal S-prime ideals, establishing foundational results crucial for understanding the S-Krull dimension. The paper also delves into properties of S-principal ideal S-domains, S-Artinian rings, and u-S-von Neumann regular rings, introducing an S-variant of the Division Algorithm. In the final section, the concept of S-height for S-prime ideals is defined, and the S-Krull dimension is analyzed through various characterizations and examples, with a particular focus on polynomial rings. We conclude with insights into the S-height of larger S-prime ideals of R[X] and their intersections with R, contributing to a deeper understanding of the S-Krull dimension. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I an ideal of R and M a finitely generated R-module with dimR(M) = d. Denote by depthR(I,M) the depth of M in I. In [C. Huneke and V. Trivedi, The height of ideals and regular sequences, Manuscr. Math. 93 (1997) 137–142], Huneke and Trivedi proved that if R is a quotient of a regular ring then there exists a finite subset ΛM of Spec(R) such that depthR(I,M) =min픭∈Λ M{depthR픭(M픭) + ht((I + 픭)/픭)}. Denote by PsuppRi(M) = {픭 ∈Spec(R)|H픭 R픭i−dim(R/픭)(M픭)≠0} the ith pseudo support of M defined by Brodmann and Sharp [On the dimension and multiplicity of local cohomology modules, Nagoya Math. J. 167 (2002) 217–233]. In this paper, we prove that if PsuppRi(M) is closed for all i ≤ d then the above formula of depthR(I,M) holds true, where ΛM =⋃0≤i≤dmin PsuppRi(M). In particular, if R is a quotient of a Cohen–Macaulay local ring then ΛM = ⋃0≤i≤dminVar(AnnR(H픪i(M))). We also give some examples to clarify the results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of m -systems. The notion of m -system is also studied from the topological point of view. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, X be an indeterminate over R, R[X] be the polynomial ring over R, R[[X]] be the power series ring over R, and Q be a principal prime ideal of R[X] with (Q ∩ R)[X] ⊊ Q. It is well known that if R is an integral domain, then R[X]Q is a DVR and R[X] has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) RQ∩R is a field, (ii) R[X]Q is a DVR, but (iii) there is a ring R such that R[X] has no principal prime ideal. We also study the maximal ideals of R[[X]] that are principal. [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]

Let (R, 픪) be a local commutative Noetherian ring and I an ideal of R. Assume that M is a finitely generated R-module of finite projective dimension p and N is a finitely generated R-module of dimension d. We prove that H픪1(H Ip+d−1(M,N)) is Artinian. Moreover, if HomR(R/I,HIp+d−1(M,N)) is finitely generated, then HIp+d−1(M,N) is an I-cofinite R-module. [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]

Let R [ [ X ] ] be the collection of power series with coefficients in a commutative ring R with identity. For a suitable function λ from ℕ 0 to ℕ , one can define a multiplication ∗ λ in R [ [ X ] ] such that together with the usual addition, R [ [ X ] ] becomes a ring that contains R as a subring. Denote this ring by R λ [ [ X ] ]. By this observation, the usual power series ring and the well-known Hurwitz series ring are the special cases of R λ [ [ X ] ] when λ (i) = 1 for all i and λ (i) = i ! for all i , respectively. In this paper, we study the Krull dimension of R λ [ [ X ] ]. We first introduce the concept of almost strong finite type (ASFT) rings and study basic properties of this type of rings. We then show that for any function λ , the Krull dimension of R λ [ [ X ] ] is at least 2 ℵ 1 if R is a non-ASFT ring, which is an analogue of the result about the Krull dimension of the usual power series ring that dim R [ [ X ] ] ≥ 2 ℵ 1 if R is a non-SFT ring. In particular, we have the Krull dimension of the Hurwitz series ring is at least 2 ℵ 1 if R is a non-ASFT ring, which gives an answer to a question of Benhissi and Koja about the infiniteness of the Krull dimension of the Hurwitz series ring. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity. The prime ideal sum graph of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this paper, we obtain the strong metric dimension of the prime ideal sum graph for various classes of Artinian nonlocal commutative rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S be a multiplicative subset of R. We say that R has S -Noetherian spectrum if for every ideal I of R , s I ⊆ J ⊆ I for some s ∈ S and some finitely generated ideal J. In this paper, we study rings with S -Noetherian spectrum. Among other things, we give a necessary and sufficient condition for Nagata's idealization R (+) M , where M is an R -module, to satisfy the S -Noetherian spectrum. We also investigate when the amalgamated algebra along an ideal has S -Noetherian spectrum. Several examples are given to illustrate the concepts and results. [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]

In this paper, we introduce the unit-Jacobson graph, which is defined by the unit elements and the elements of the Jacobson radical of a commutative ring R with nonzero identity. We give relationships between this new graph concept and some special rings such as j-clean rings, UJ-rings, local rings, and cartesian rings. Moreover, we investigate the concepts of the dominating set, diameter, and girth on the unit-Jacobson graph. [ABSTRACT FROM AUTHOR]

Let λ = λ0 + uλ1 + ⋯ + uk−1λ k−1 be a Type 1 unit in ℜk = 픽pm + u픽pm + ⋯ + uk−1픽 pm(uk = 0), where p is an odd prime, m is a positive integer and λ0,λ1,…,λk−1 ∈ 픽pm,λ0≠0,λ1≠0. In this paper, we give the complete structure of all Type 1 λ-constacyclic codes and their duals of length 8ps over the finite commutative chain ring ℜk in terms of their generator polynomials. Using this structure, we determine the Hamming distance and the Rosenbloom–Tsfasman (RT) distance of all Type 1 λ-constacyclic codes. For pm ≡ 1(mod 4) and a unit λ ∈ ℜ2, we determine the b -symbol distances of all λ-constacyclic codes of length 8ps over ℜ2, where b ≤ 8. As illustrations, we provide several λ-constacyclic codes with new parameters with respect to Hamming, RT and b-symbol metrics. MDS codes are widely recognized for their optimal error-correction capability, and MDS b-symbol codes are generalization of MDS codes. We found some MDS b-symbol constacyclic codes of length 8ps over ℜ2. Additionally, for pm ≡ 1(mod 4), we provide a decoding algorithm for Type 1 constacyclic codes of length 8ps over ℜk with respect to the Hamming, RT and b-symbol metrics. [ABSTRACT FROM AUTHOR]

This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension. [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 (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]

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]

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]

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]

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 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]

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

In this paper, we continue the study of Hurwitz series over a commutative unital ring that was begun in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. In particular, we introduce the notion of multivariate Hurwitz series. The underlying idea is that multivariate Hurwitz series are to Hurwitz series as studied in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] as formal power series in several indeterminates are to formal power series in only one indeterminate. The elementary aspects of the theory follow along the lines of [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. The treatment of substitution and divided powers introduces special problems not encountered in [W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] and requires special attention to subtle details. However, we are able to establish analogous results. With substitution and divided powers in place, we construct and study the analog to the so called inner transformations of [S. Bochner and W. T. Martin, Several Complex Variables (Princeton University Press, 1948)]. Finally, we are able to establish analogs to many of the fundamental results of single and multivariate calculus. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. A proper ideal I of R is called a 1-absorbing prime ideal (respectively, 1-absorbing primary ideal) if whenever nonunit elements a , b , c ∈ R with a b c ∈ I , then a b ∈ I or c ∈ I (respectively, a b ∈ I or c ∈ I ). The purpose of this paper is to study the transfer of certain 1-absorbing-like properties to amalgamation of A with B along J with respect to f (denoted by A ⋈ f J), introduced and studied by D'Anna, Finocchiaro and Fontana. Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties. [ABSTRACT FROM AUTHOR]

Recall that a ring R is a Hilbert ring if any maximal ideal of R [ X ] contracts to a maximal ideal of R. The main purpose of this paper is to characterize the prime ideals of a commutative ring R which are traces of the maximal ideals of the polynomial ring R [ X ]. In this context, we prove that if p is a prime ideal of R such that R / p is a semi-local domain of (Krull) dimension ≤ 1 , then p is the trace of a maximal ideal of R [ X ]. Whereas, if R is Noetherian and either (dim (R / p) ≥ 2) or (the quotient field of R / p is algebraically closed, dim (R / p) = 1 and R / p is not semi-local), then p is never the trace of a maximal ideal of R [ X ]. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings R [ X ] and the finiteness of the Ext-index of localizations of R. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. The co-maximal ideal graph of R , denoted by Γ (R) , is a simple graph whose vertices are proper ideals of R which are not contained in the Jacobson radical J (R) of R and two distinct vertices I , J are adjacent if and only if I + J = R. In this paper, we use Gallai's Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established. [ABSTRACT FROM AUTHOR]

This paper shows that if R is a homomorphic image of a Cohen–Macaulay local ring, then R -module M is sequentially generalized Cohen–Macaulay if and only if the difference between Hilbert coefficients and arithmetic degrees for all distinguished parameter ideals of M are bounded. [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]

In this paper, we introduce weakly 1-absorbing primary submodules of modules over commutative rings. Let R be a commutative ring with a nonzero identity and M be a nonzero unital module. A proper submodule N of M is said to be a weakly 1-absorbing primary submodule if whenever 0 ≠ a b m ∈ N for some nonunit elements a , b ∈ R and m ∈ M , then a b ∈ (N : M) or m ∈ M - rad (N) , where M - rad (N) is the prime radical of N. Many properties and characterizations of weakly 1-absorbing primary submodules are given. We also give the relations between weakly 1-absorbing primary submodules and other classical submodules such as weakly prime, weakly primary, weakly 2-absorbing primary submodules. Also, we use them to characterize simple modules. [ABSTRACT FROM AUTHOR]

Let (, ,) be an Ext -finite, Krull–Schmidt and k -linear n -exangulated category with k a commutative artinian ring. In this note, we prove that has Auslander–Reiten–Serre duality if and only if has Auslander–Reiten n -exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander–Reiten–Serre duality). Finally, assume further that has Auslander–Reiten–Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander–Reiten–Serre duality. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Krull dimension-dependent elements of a Noetherian commutative ring. Let x , y be non-unit elements of a commutative ring R. x , y are called Krull dimension-dependent elements, whenever dim R / (R x + R y) = min { dim R / R x , dim R / R y }. We investigate the elements of a ring according to this property. Among the many results, we characterize the rings that all elements of them are Krull dimension-dependent and we call them, closed under the Krull dimension. Moreover, we determine the structure of the rings with Krull dimension at most 1, that are closed under the Krull dimension. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of r -ideals. A proper ideal I of R is said to be a (1 , r) -ideal if whenever nonunit elements a , b , c ∈ R and a b c ∈ I , then a b ∈ I or c ∈ Z (R). Some basic properties of (1 , r) -ideals are studied. For instance, we give a method to construct (1 , r) -ideals that are not r -ideals. Among other things, it is shown that if R admits a (1 , r) -ideal that is not an r -ideal, then R is a local ring. Also, we provide a necessary and sufficient condition in term of (1 , r) -ideals for a ring to be a total quotient ring and we determinate the (1 , r) -ideals of a chained ring. Finally, we give an idea about some (1 , r) -ideals of the localization of rings, the trivial ring extensions and the power series rings to construct nontrivial and original examples of (1 , r) -ideals. [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]

A function f : R → R , where R is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative p ∈ R [ x ]. Based on this notion, we introduce ring invariants which associate to R the numbers s (R) and s (R ′ ; R) , where R ′ is the subring generated by 1. For the ring R = ℤ / n ℤ the invariant s (R) coincides with the number theoretic Smarandache or Kempner function s (n). If every function in a ring R is a polyfunction, then R is a finite field according to the Rédei–Szele theorem, and it holds that s (R) = | R |. However, the condition s (R) = | R | does not imply that every function f : R → R is a polyfunction. We classify all finite commutative rings R with unit element which satisfy s (R) = | R |. For infinite rings R , we obtain a bound on the cardinality of the subring R ′ and for s (R ′ ; R) in terms of s (R). In particular we show that | R ′ | ≤ s (R) !. We also give two new proofs for the Rédei–Szele theorem which are based on our results. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and I be an ideal of R such that the R -modules H I i (M) are I -cofinite, for all finitely generated R -modules M and all i ∈ ℕ 0 . In this paper, we investigate a question of Hartshorne concerning the Abelianness of the category of all I -cofinite modules. [ABSTRACT FROM AUTHOR]

A commutative ring R has the unique decomposition into ideals (UDI) property if, for any R -module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In [P. Goeters and B. Olberding, Unique decomposition into ideals for Noetherian domains, J. Pure Appl. Algebra 165 (2001) 169–182], the UDI property has been characterized for Noetherian integral domains. In this paper, we aim to study the UDI-like property for strong Mori domains; domains satisfying the ascending chain condition on w -ideals. [ABSTRACT FROM AUTHOR]

For a commutative ring R denote the Jacobson radical of the ring by J (R). Khashan et al. introduced and studied J-ideals and weakly J-ideals for commutative rings with identity. In this paper, let R be a non-commutative ring. We introduce weakly ρ -ideals for a special radical ρ and show that if ρ is the Jacobson radical many of the results proved by Khashan et al. are also satisfied for non-commutative rings. The results for weakly J-ideals for non-commutative rings follow as special cases of a more general situation. [ABSTRACT FROM AUTHOR]