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AbstractLet R be a commutative ring with identity and R′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R′ is finitely generated, i.e., R′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R≤2 , then R′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R=∞ or reg-dim R=n≤ dim R=n+m−1 for arbitrary positive integers n, m. [ABSTRACT FROM AUTHOR]

A quasi-order on a set S is a binary, reflexive and transitive relation on S. In [3], Fakhruddin introduced the notion of (totally) quasi-ordered fields and showed that each such field is either an ordered field or else a valued field. Hence, quasi-ordered fields are very well suited to treat ordered and valued fields simultaneously. The aim of the present paper is to prove that an analogous dichotomy holds for commutative rings with 1 as well. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity 1 ≠ 0 and T be the ring of all n × n upper triangular matrices over R. In this paper, we describe the zero divisor graphof T. Some basic graph theory properties ofare given, including determination of the girth and diameter. The structure ofis discussed, and bounds for the number of edges are given. In the case that R is a finite integral domain and n = 2, the structure ofis fully described and an explicit formula for the number of edges is given. [ABSTRACT FROM PUBLISHER]

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ℳ of the form ℳ = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided. [ABSTRACT FROM AUTHOR]

The ring of integer-valued polynomials on an arbitrary integral domain is well studied. In this paper, we initiate and provide motivation for the study of integer-valued polynomials on commutative rings and modules. Several examples are computed, including the integer-valued polynomials over the ring for any commutative ring R and any elements of R, as well as the integer-valued polynomials over the Nagata idealization R(+)M of M over R, where M is an R-module. These examples are motivated by and provide motivation for the study of integer-valued derivatives. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic p>0 and let KG be the group algebra of an arbitrary group G over K. It is known that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least p+1. The group algebras KG for which these indices are p+1 or 2p or 3p−1 or 4p−2 have already been determined. In this paper, we classify the group algebras KG for which the upper Lie nilpotency index is 5p−3, 6p−4 or 7p−5. [ABSTRACT FROM AUTHOR]

Letℛbe a commutative ring with identity and let𝔄 = Tri(𝒜,ℳ,ℬ) be a triangular algebra consisting of unital algebras𝒜,ℬoverℛand an (𝒜,ℬ)-bimoduleℳwhich is faithful as a left𝒜-module as well as a rightℬ-module. In this paper, we prove that under certain assumptions every nonlinear generalized Lie triple derivationGL:𝔄→𝔄is of the formGL = δ+τ, whereδ:𝔄→𝔄is an additive generalized derivation on𝔄andτis a mapping from𝔄into its center which annihilates all Lie triple products [[x,y],z]. [ABSTRACT FROM AUTHOR]

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2] and were extensively investigated by Ahn and Anderson [1], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings. [ABSTRACT FROM AUTHOR]

This paper introduces the notions of a zero-divisor labeling and the zero-divisor index of a graph using the zero-divisors of a commutative ring. Viewed in this way, the usual zero-divisor graph is a maximal graph with respect to a zero-divisor labeling. We also study optimal zero-divisor labelings of a finite graph. [ABSTRACT FROM PUBLISHER]

LetAbe a differential graded (DG) algebra with a trivial differential over a commutative unital ring. This paper investigates the image of the totaling functor, defined from the category of complexes of gradedA-modules to the category of DGA-modules. Specifically, we exhibit a special class of semifree DGA-modules which can always be expressed as the totaling of some complex of graded freeA-modules. As a corollary, we also provide results concerning the image of the totaling functor whenAis a polynomial ring over a field. [ABSTRACT FROM AUTHOR]

This paper investigates ideal-theoretic as well as homological extensions of the Prüfer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prüfer conditions. [ABSTRACT FROM AUTHOR]

LetRbe a commutative ring andManR-module. The purpose of this paper is to investigate the setT(M) = {m ∈ M | rm = 0 for some 0 ≠ r ∈ R} of torsion elements ofM. Of particular interest is the question of whenT(M) is a submodule ofM. We show that ifM ≠ T(M), thenT(M) is a union of prime submodules ofM. [ABSTRACT FROM PUBLISHER]

The notion of (FC)-sequences was built in Viet [Viet, D. Q. (2000). Mixed multiplicities of arbitrary ideals in local rings. Comm. Algebra 28(8):3803–3821] containing important information for multiplicities and reductions of ideals [see Viet, D. Q. 2000; Viet, D. Q. (2003a). On some properties of (FC)-sequences of ideals in local rings. Proc. Am. Math. Soc. 131:45–53; Viet, D. Q. (2003b). Sequences determining mixed multiplicities and reductions of ideals. Comm. Algebra 31(10):5047–5069; Viet, D. Q. (2003c). On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple. J. Pure Appl. Algebra 183:313–327]. In this paper, we will construct generalized joint reductions generated by maximal weak-(FC)-sequences of a set of arbitrary ideals. We showed that the mixed multiplicity of ideals and the mixed multiplicity of their reductions are the same. The paper also determines the vanishing and non-vanishing of mixed multiplicities and as an application of above results we compute mixed multiplicities and the multiplicity of multi-graded Rees algebras in some particular cases. [ABSTRACT FROM AUTHOR]

A ring R is called clean if every element of it is a sum of an idempotent and a unit. A ring R is neat if every proper homomorphic image of R is clean. When R is a field, then a complete characterization has been obtained for a commutative group ring RG to be neat, but not clean. And if R is not a field, then necessary conditions are obtained for a commutative group ring RG to be neat, but not clean. A counterexample is given to show that these necessary conditions are not sufficient. [ABSTRACT FROM AUTHOR]

Recall that a commutative ring R is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module M is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules. [ABSTRACT FROM AUTHOR]

We study the ring of polyfunctions over Z / n Z. The ring of polyfunctions over a commutative ring R with unit element is the ring of functions f : R → R which admit a polynomial representative p ∈ R [ x ] in the sense that f (x) = p (x) for all x ∈ R. This allows to define a ring invariant s which associates to a commutative ring R with unit element a value in N ∪ { ∞ }. The function s generalizes the number theoretic Smarandache function. For the ring R = Z / n Z we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number Ψ (n) of polyfunctions over Z / n Z. We also investigate algebraic properties of the ring of polyfunctions over Z / n Z. In particular, we identify the additive subgroup of the ring and the ring structure itself. Moreover we derive formulas for the size of the ring of polyfunctions in several variables over Z / n Z , and we compute the number of polyfunctions which are units of the ring. [ABSTRACT FROM AUTHOR]

We determine the number of edges of the finite direct product of finite rings. We apply this result to finite rings without idempotents, in particular direct products of ℤm. [ABSTRACT FROM PUBLISHER]

We will give quiver presentations of the Grothendieck constructions of functors from a small category to the 2-category of 𝕜-categories for a commutative ring 𝕜. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the study of the Krull dimension of tensor products of algebras over a field k. More precisely, we seek satisfactory analogues of Seidenberg's inequalities of polynomial rings for tensor products of k-algebras. As an application, we supply a new range of k-algebras A and B for which the Krull dimension of A ⊗ k B may be computed. [ABSTRACT FROM AUTHOR]

A right ideal I of a ring R is said to be a comparizer right ideal of R if for any right ideals A , B of R , either A ? B or BI ? A . A ring R (with 1) is called a right pseudo-chain ring if for any nonunit a ? R , aR is a comparizer right ideal of R . These rings are generalizations of right chain rings and commutative pseudo-valuation rings. In this paper we study the structure of right pseudo-chain rings. We prove that a ring R containing a non-zero-divisor in the Jacobson radical J ( R ) is a pseudo-chain ring if and only if R is a local subring of a (unique up to isomorphism) chain ring T with J ( T ) = J ( R ). We also introduce the concept of a pseudo-uniserial module. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce the concept of integrally closed modules and characterize Dedekind modules and Dedekind domains. They also show that a given domain R is integrally closed if and only if a finitely generated torsion-free projective R -module is integrally closed. In addition, it is proved that any invertible submodule of a finitely generated projective module over a domain is finitely generated and projective. Also they give the equivalent conditions for Dedekind modules and Dedekind domains. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism s ? Aut( R ), we use the notion of a s-compatible module M R to obtain results on the couniform dimension of the polynomial modules M [ x ], M [ x -1 ], and M [ x , x -1 ] over suitable skew extension rings. [ABSTRACT FROM AUTHOR]

Given a family of commutative rings { R a } a? A , we investigate the family ??(? a? A R a ) of Artinian subrings of ? a? A R a . This paper deals also with the question of when a pair (? a? A R a , ? a? A T a ) is zero-dimensional, where {( R a , T a )} a? A is a family of zero-dimensional pairs. [ABSTRACT FROM AUTHOR]

Let A be a noetherian local ring with dimension d and I be an ideal of A . Let E=( E n ) n = 0 be a good I -filtration of submodules of an A -module E . Let H be an ideal of A containing I and F H E) = ? n = 0 E n / HE n . Assume that E is a Cohen–Macaulay module with ?( E / IE ) finite and Ann ( E ) = 0, and let J be a minimal reduction of I . In this paper we give conditions on ?( E n n JE / JE n -1 ) and ?( HE n n JE / JHE n -1 ) so, that F H (E), has depth of at least d - 1. [ABSTRACT FROM AUTHOR]

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E* ⊗ G induces the Koszul complex ...→ SmE* ⊗ SnG ⊗ [This symbol cannot be presented in ASCII format] P (E* ⊗ G) → Sm+1E* ⊗ Sn+1 G ⊗ [This symbol cannot be presented in ASCII format] P-1 (E* ⊗ G) → ... and its dual ... → Dm+1 E ⊗ D n+1 G* ⊗ [This symbol cannot be presented in ASCII format]P-1 (E⊗G*) → DmE ⊗ DnG* ⊗ [This symbol cannot be presented in ASCII format] (E⊗G*) → ... Let HN(m,n,p) and HM(m,n,p) be the homology of the above complexes at SmE* ⊗ SnG ⊗ [This symbol cannot be presented in ASCII format] P (E* ⊗ G) and DmE ⊗ Dn G* ⊗ [This symbol cannot be presented in ASCII format] P (E ⊗G*), respectively. In this paper, we investigate the modules HN(m,n,p) and HM(m,n,p) when -e ≤ m - n ≤ g. We record the fact, already implicitly calculated by Bruns and Guerrieri, that HN(m,n,p) ≅ H M (m′,n′,p′), provided m + m′ = g - 1, n + n′ = e-1, p+p′ = (e-1)(g-1), and 1-e ≤ m - n ≤ g - 1. If m - n is equal to either g or -e, then we prove that the only nonzero modules of the form H N(m,n,p) and HM(m,n,p) appear in one of the split exact sequences. 0 → H M(g,0,p′) → [This symbol cannot be presented in ASCII format] g+p′ (E ⊗ G*) → H N(0,e,p) → 0, or 0 → H M(0,e,p′) → [This symbol cannot be presented in ASCII format] e+p′ (E ⊗ G*) → H N(g,0,p) → 0, where p + p′ = (e - 1)(g - 1) -1. The modules that we study are not always free modules. Indeed, if m=n, then the module H N(m,n,p) is equal to a homogeneous summand of the graded module Tor[This symbol cannot be presented in ASCII format](T,R), where P is a polynomial ring in eg variables over R and T is the determinantal ring defined by the 2 x 2 minors of the corresponding e x g matrix of indeterminates. Hashimoto's work shows that if e and g are both at least five, then H N(2,2,3) is not a free module when R is Z, and when R is a field, the rank of this module depends on the characteristic of R. When the modules H M (m,n,p) are free, they are summands of the resolution of the universal ring for finite length modules of projective dimension two. [ABSTRACT FROM AUTHOR]

In this paper, we study codivisorial injective modules over a ring of Krull type. Essentially, we obtain two main results: First, we give a necessary and sufficient condition for a ring of Krull type A to be independent ring of Krull type. Second, we prove that any direct sum of codivisorial injective modules isξ-injective; whereξ is a set of some special ideals. [ABSTRACT FROM AUTHOR]

In the derived category of the category of modules over a commutative Noetherian ring R, we define, for an ideal [afr] of R, we investigate the interplay between the two naturally defined numbers of cohomological dimensions of a complex X in a certain subcategory of the derived category, namely cd([afr], X) = sup{cd([afr], Hℓ(X)) − ℓ|ℓ ∈ [Zopf]} and− infRΓ[afr](X). Here cd([afr], M) = sup{ℓ ∈ [Zopf]| for an R-module M, and infRΓ[afr](X) denotes the homological infimum of the complex RΓ[afr](X). In this paper, it is shown, among other things, that, for any complex X bounded to the left,− infRΓ[afr](X) ≤ cd([afr], X) and equality holds if indeed H(X) is finitely generated. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M) = {PMP ∈ Spec(R) and P ⊇ M⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal. [ABSTRACT FROM AUTHOR]

In this paper we extend a well-known characterization of unital Artinian serial rings to the analagous class of rings with local units. In particular we characterize locally Artinian serial rings as those in which every finitely generated module or every finitely cogenerated module is uniserial. Additionally, we show that the entire characterization does not extend, as there exist locally Artinian serial rings that have non-serial modules. [ABSTRACT FROM AUTHOR]

The zero-divisor graphs of commutative rings have been used to build bridges between ring theory and graph theory. Namely, they have been used to characterize many ring properties in terms of graphic ones. However, many results are established only for reduced rings because a zero-divisor graph defined in the classical manner lacks the information on relationship between powers of zero-divisors. The aim of this article is to remedy this situation by introducing a parametrized family of graphs { Γ ¯ i (R) } i ∈ N * , for a ring R, which reveals more of the relationship between powers of zero-divisors as follows: For each i ∈ N * , Γ ¯ i (R) is the simple graph whose vertex set is the set of non-zero zero-divisors such that two distinct vertices x and y are joined by an edge if there exist two positive integers n ≤ i and m ≤ i such that x n y m = 0 with x n ≠ 0 and y m ≠ 0. Our aim is to study in detail the behavior of the filtration { Γ ¯ i (R) } i ∈ N * as well as the relations between its terms. We give answers to several interesting and natural questions that arise in this context. In particular, we characterize girth and diameter of Γ ¯ i (R) and give various examples. [ABSTRACT FROM AUTHOR]

A commutative ring R is said to satisfy property (P) if every finitely generated proper ideal of R admits a non-zero annihilator. In this paper we give some necessary and sufficient conditions that a ring satisfies property (P). In particular, we characterize coherent rings, noetherian rings and Π-coherent rings with property (P). [ABSTRACT FROM AUTHOR]

Let p be a prime number. Let B be the idealization ℤ(+)ℤ∕pℤ and N: = pℤ(+)ℤ∕pℤ. Let B⊂R be a ramified (integral minimal) ring extension, with crucial maximal ideal 풩 and (necessarily) an element y such that R = B[y], y2∈B, y3∈B and y풩⊆풩. Then B is the only (commutative unital) ring properly contained between ℤ and R if and only if 풩 = N and either or py∉ℤ. Moreover, there are exactly three isomorphism classes of such rings R. Let A: = ℤ∕pαℤ for some integer α≥2. Let A⊂D⊂E be a tower of ramified ring extensions, (necessarily) with an element z such that E = D[z], z2∈D, z3∈D and zℳ⊆ℳ, where ℳ is the crucial maximal ideal of D⊂E. Then D is the only ring properly contained between A and E if and only if either z2∉A or pz∉A. There are at least two, but only finitely many, isomorphism classes of such rings E. We complete the characterization of the rings with exactly two proper (unital) subrings, including a classification up to isomorphism of these rings in the case of characteristic 0. [ABSTRACT FROM AUTHOR]

A ring is called an r-Noetherian ring if every regular ideal is finitely generated. Let R be an r-Noetherian ring, let I be a regular ideal of R, and let be the I-adic completion of R. We show that is a Noetherian ring and dim  =  sup{r-ht(M)∣M∈Max(R) and I⊆M}. Let P be a prime ideal of R. We also prove that for any a∈reg(P), r-htP = ht(P∕aR)+1 and that if P is minimal over an n-generated regular ideal, then r-htP≤n. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring with center Z(R) and with extended centroid C. Suppose that τ:R→R is an anti-homomorphism such that the image of τ has small centralizer. It is proved that the following are equivalent: (1) for all x∈R; (2) x+xτ∈Z(R) for all x∈R; (3) xxτ∈Z(R) for all x∈R. In this case, there exists an idempotent e∈C such that (1−e)R is a commutative ring and the semiprime ring eR is equipped with an involution , which is induced canonically by τ. These results generalize both the involution case by Chacron [5] and the anti-automorphism case by the author [7]. [ABSTRACT FROM AUTHOR]

A commutative ringRis J-stable provided thatR∕aRhas stable range 1 for alla∉J(R). A commutative ringRin which every finitely generated ideal principal is called a Bézout ring. A ringRis an elementary divisor ring provided that every matrix overRadmits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g. [3, Theorem 8], [4, Theorem 4.1], [7, Theorem 3.7], [8, Theorem], [9, Theorem 2.1], [14, Theorem 1] and [18, Theorem 7]. [ABSTRACT FROM AUTHOR]

Kuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihilator” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings. [ABSTRACT FROM AUTHOR]

A ringRis calledleft fusibleif every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor. It is shown that ifRis a left fusible ring andσis a ring automorphism ofR, thenR[x;σ] andR[[x;σ]] are left fusible. It is proved that ifRis a left fusible ring, thenMn(R) is a left fusible ring. Examples of fusible rings are complemented rings, special almost clean rings, and commutative Jacobson semisimple clean rings. A ringRis calledleft unit fusibleif every nonzero element ofRcan be written as the sum of a unit and a left zero-divisor inR. Full rings of continuous functions are fusible. It is also shown that ifin a ringR, where theeiare orthogonal idempotents and eachis left unit fusible, thenRis left unit fusible. Finally, we give some properties of fusible rings. [ABSTRACT FROM PUBLISHER]

For certain classes 𝒞 ofR-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/Mis Noetherian for any finitely generated moduleMR. RingsRwith latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. IfRis a quasi continuous ring in 𝒬 thenR ≃ A ⊕ TwhereAis a semisimple Artinian ring andT ∈ 𝒬 with Z(TT) ≤essTT. [ABSTRACT FROM PUBLISHER]

In this article, we introduce the dual notion of content (i.e., cocontent) modules over commutative rings and investigate some basic properties of this class of modules. [ABSTRACT FROM PUBLISHER]

Let S be a commutative ring with identity and R a unitary subring of S. An ideal I of S is called an R-conductor ideal of S if I = {x ∊ S | xS ⊆ V} for some intermediate ring V of R and S. In this note, we present necessary and sufficient criteria for being an R-conductor ideal of S. We generalize several well known facts about them and present a simple approach to rediscover the results of both old and recent articles. We sketch the boundaries of our criteria by providing a few counterexamples. [ABSTRACT FROM AUTHOR]

LetRbe a commutative ring,Q0(R) be the ring of finite fractions overR, andwbe the so-calledw-operation onR. In this article, we introduce a new type of Prüferv-multiplication ring, called a quasi-Q0-PvMR and defined as a ringRfor which everyw-linkedQ0-overring ofRis integrally closed inQ0(R). Our primary motivation for investigating quasi-Q0-PvMRs is to providew-theoretic analogues to some work of Lucas [16] concerningQ0-Prüfer rings. (A ringRis called aQ0-Prüfer ring if everyQ0-overring ofRis integrally closed inQ0(R).) [ABSTRACT FROM PUBLISHER]

The content of a polynomialfover a commutative ringRis the idealc(f) ofRgenerated by the coefficients off. A commutative ringRis said to be Gaussian ifc(fg) = c(f)c(g) for every polynomialsfandginR[X]. A number of authors have formulated necessary and sufficient conditions forR(X) (respectively,R⟨X⟩) to be semihereditary, have weak global dimension at most one, be arithmetical, or be Prüfer. An open question raised by Glaz is to formulate necessary and sufficient conditions thatR(X) (respectively,R⟨X⟩) have the Gaussian property. We give a necessary and sufficient condition for the ringsR(X) andR⟨X⟩ in terms of the ringRin case the square of the nilradical ofRis zero. [ABSTRACT FROM PUBLISHER]

Let A be a commutative ring andEa nonzeroA-module. Necessary and sufficient conditions are given for the trivial ring extensionRofAbyEto be either arithmetical or Gaussian. The possibility forRto be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqp-ring. Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1, or ∞. [ABSTRACT FROM PUBLISHER]

LetRbe a commutative Noetherian ring with nonzero identity, 𝔞 an ideal ofR,Ma finiteR–module,Xan arbitraryR–module, andna non-negative integer. Here, we show that, in the Serre subcategories of the category ofR–modules, how the generalized local cohomology modules, the ordinary local cohomology modules, and the extension modules behave similarly at the initial pointsi ≤ n. We conclude with some Artinianness and cofiniteness results for, and some finiteness results forand. [ABSTRACT FROM AUTHOR]

LetLbe a finite-dimensional complex simple Lie algebra,Lℤbe the ℤ-span of a Chevalley basis ofL, andLR = R ⊗ℤLℤbe a Chevalley algebra of typeLover a commutative ringR. Let 𝒩(R) be the nilpotent subalgebra ofLRspanned by the root vectors associated with positive roots. A map ϕ of 𝒩(R) is called commuting if [ϕ(x),x] = 0 for allx∈ 𝒩(R). In this article, we prove that under some conditions forR, if Φ is not of typeA2, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of typeA2, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism). [ABSTRACT FROM AUTHOR]

This note is concerned with generalizations of commutativity. We introduceidentity-symmetricandright near-commutative, and study basic structures of rings with such ring properties. It is shown that ifRis an identity-symmetric ring, then the set of all nilpotent elements forms a commutative subring ofR. Moreover, identity-symmetric regular rings are proved to be commutative. The near-commutativity is shown to be not left-right symmetric, and we study some conditions under which the near-commutativity is left-right symmetric. We also examine the near-commutativity of skew-trivial extensions, which has a role in this note. [ABSTRACT FROM AUTHOR]

LetFbe an affine flat group scheme over a commutative ringR, andSanF-algebra (anR-algebra on whichFacts). We define an equivariant analogueQF(S) of the total ring of fractionsQ(S) ofS. It is the largestF-algebraTsuch thatS ⊂ T ⊂ Q(S), andSis anF-subalgebra ofT. We study some basic properties. Utilizing this machinery, we give some new criteria for factoriality (unique factorization domain property) of (semi-)invariant subrings under the action of affine algebraic groups, generalizing a result of Popov. We also prove some variations of classical results on factoriality of (semi-)invariant subrings. Some results over an algebraically closed base field are generalized to those over an arbitrary base field. [ABSTRACT FROM AUTHOR]

LetRbe a commutative ring andX = RgMax(R) be the set of all maximal subrings ofR. We give a topology onXby putting 𝕏(S) = {T ∈ X | S ⊆ T}, whereSranges over all subrings ofR, as a subbase for closed subsets forX. We investigate the decomposition into irreducible components for this topology. It is shown that valuation domains behave similar to prime ideals in Zariski topology in our topology. Further we present an analogous form of the Prime Avoidance Lemma for valuation domains instead of prime ideals. The compactness of 𝕏(S) for certain subringsSofRis determined. Moreover, we characterize fieldsEfor which the spaceX = RgMax(E) is compact. [ABSTRACT FROM AUTHOR]

In this work, we propose a method for computing noncommutative Gröbner bases over a noetherian valuation ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical commutative Gröbner bases is generalized to Buchberger's algorithm overR= 𝒱 ⟨ x1,…,xm ⟩, a free associative algebra with noncommuting variables, where 𝒱 = ℤ/nℤ and 𝒱 = ℤ. The proposed process generalizes previous known techniques for the computation of commutative Gröbner bases over a nætherian valuation ring and/or noncommutative Gröbner bases over a field. [ABSTRACT FROM AUTHOR]