Prüfer property in amalgamated algebras along an ideal

Let $$f : A \rightarrow B$$ be a ring homomorphism and J be an ideal of B. In this paper, we give a characterization of zero divisors of the amalgamation which is a generalization of Maimani’s and Yassemi’s work (see Maimani and Yassemi in J Pure Appl Algebra 212(1):168–174, 2008). Furthermore, we investigate the transfer of Prüfer domain concept to commutative rings with zero divisors in the amalgamation of A with B along J with respect to f (denoted by $$A\bowtie ^fJ),$$ introduced and studied by D’Anna et al. (Commutative algebra and its applications, Walter de Gruyter, Berlin, 2009, J Pure Appl Algebra 214:1633–1641, 2010). Our results recover well known results on duplications. The main applications constist in the construction of new original classes of Prüfer rings that are not Gaussian and Prüfer rings with weak global dimension strictly greater than 1.     

Sufficient Condition to Resolve Costa's First Conjecture

In this work, we give a sufficient condition to resolve Costa's first conjecture for each positive integer n and d with n ≥ 4. Precisely, we show that if there exists a local ring (A, M) such that λ _A (M) = n, and if there exists an (n + 2)-presented A-submodule of M ^m , where m is a positive integer (for instance, if M contains a regular element), then we may construct an example of (n + 4, d)-ring which is neither an (n + 3, d)-ring nor an (n + 4, d ? 1)-ring. Finally, we construct a local ring (B, M) such that λ _B (M) = 0 (resp., λ _B (M) = 1) and so we exhibit for each positive integer d, an example of a (4, d)-ring (resp., (5, d)-ring) which is neither a (4, d ? 1)-ring (resp., neither a (5, d ? 1)-ring) nor a (2, d′)-ring (resp., nor a (3, d′)-ring) for each positive integer d′.     

On FF-rings

In this paper, we study the class of rings in which every flat ideal is finitely generated. We investigate the stability of this property under localization and homomorphic image, and its transfer to various contexts of constructions such as direct products, pullback rings, and trivial ring extensions. Our results generate examples which enrich the current literature with new and original families of non-Noetherian rings that satisfy this property.     

On (Strongly) Gorenstein Von Neumann Regular Rings

In this article, we introduce and study the rings over which every module is (strongly) Gorenstein flat, which we call them (strongly) Gorenstein Von Neumann regular rings.     

On Strong (A)-Rings

In this paper, we introduce a strong property (A) as follows: A ring R is called satisfying strong property (A) if every finitely generated ideal of R which is generated by a finite number of zero-divisors elements of R, has a non zero annihilator. We study the transfer of property (A) and strong property (A) in trivial ring extensions and amalgamated duplication of a ring along an ideal. We also exhibit a class of rings which satisfy property (A) and do not satisfy strong property (A).     

On Almost Valuation and Almost Bézout Rings

In this paper we extend the notion of almost valuation and almost Bézout domains to arbitrary commutative rings, and we investigate the transfer of these properties to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide new classes of commutative rings satisfying these properties. As an immediate consequence, we show the failure of Anderson–Zafrullah's theorem on the integral closure of an almost valuation domain beyond the context of integral domains.     

Gorenstein Global Dimension of an Amalgamated Duplication of a Coherent Ring Along an Ideal

In this paper, we study the Gorenstein global dimension of an amalgamated duplication of a coherent ring along a regular principal ideal.     

Trivial Ring Extensions Defined by Arithmetical-Like Properties

In this article we investigate the transfer of the notions of elementary divisor ring, Hermite ring, Bezout ring, and arithmetical ring to trivial ring extensions of commutative rings by modules. Namely, we prove that the trivial ring extension R: = A ? B defined by extension of integral domains is an elementary divisor ring if and only if A is an elementary divisor ring and B = qf(A); and R is an Hermite ring if and only if R is a Bezout ring if and only if A is a Bezout domain and qf(A) = B. We provide necessary and sufficient conditions for R = A ? E to be an arithmetical ring when E is a nontorsion or a finitely generated A ? module. As an immediate consequences, we show that A ? A is an arithmetical ring if and only if A is a von Neumann regular ring, and A ? Q(A) is an arithmetical ring if and only if A is a semihereditary ring.     

The ( n,d )-Krull Property over Domains Arising from Pullbacks

In this work we investigate the transfer of the (n,d)-Krull property between domains arising from pullbacks. As an application, we construct a new class of non-coherent domains which are (0,d)-Krull domains, not (0,d ? 1)-Krull domains, and where the Krull dimension equals d + 1.     

Graded pseudo-valuation domains

Abstract

We introduce a graded version of pseudo-valuation rings for integral domains graded by a torsionless monoid, and we extend several results to the graded situation.