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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]

The main purpose of this paper is to totally characterize when the amalgamated duplication R ⋈ I of a ring R along an ideal I is an -ring as well as an -ring. In this regard, we prove that R ⋈ I is an -ring if and only if R is an -ring and I is contained in the set of zero divisors Z(R) of R. As to the Property () of R ⋈ I, it turns out that its characterization involves a new concept that we introduce in [6] and that we term the Property () of a module M along an ideal I. In fact, we prove that R ⋈ I is an -ring if and only if R is an -ring, I is an -module along itself and if p is a prime ideal of R such that p ⊆ ZR(I) ∪ Z1 (R), then either p ⊆ ZR(I) or p ⊆ Z1 (R), where Z1 (R) := {a ∈ R : a + I ⊆ Z1z(R)}. [ABSTRACT FROM AUTHOR]

In this paper we show that if {Ri}i∈I is a family of commutative rings and {Mi}i∈I is a family of modules such that each Mi is an Ri-module, then the direct product Mi is a -module over Ri if and only if each Mi is an -module over Ri, i ∈ I. This result extends the work of Hong, Kim, Lee and Ryu in Rings with Property () and their extensions, J. Algebra315 (2007), 612–628. Moreover, we characterize Property () for modules in terms of Property () of their total quotient modules, extending the work of Dobbs and Shapiro in On the strong ()-ring of Mahdou and Hassani, Mediterr. J. Math.10 (2013), 1995–1997. [ABSTRACT FROM AUTHOR]

We present an answer to a problem raised by Anderson and Chun in [D. D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16(7) (2017) 1750143] on characterizing when the idealization R ⋉ M of a ring R on an R -module M is an 𝒜 -ring (respectively, an 𝒮 𝒜 -ring) in terms of module-theoretic properties of R and M. Also, we are concerned with an open question asked by these two authors which reads the following: What modules over a given ring R are homomorphic images of modules satisfying the strong Property 𝒜 ? (see, Question 4.4(1) in the above mentioned paper). This paper highly contributes to answer such a question. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to generalize a result of T.G. Lucas which states that if R is a reduced commutative ring and M is a flat R-module, then the idealization R ⋉ M is an A -ring if and only if R is an A -ring [14, Proposition 3.5]. In effect, we drop the reduceness hypotheses and prove that, given an arbitrary commutative ring R and any submodule M of a flat R-module F, R ⋉ M is an A -ring (resp., S A -ring) if and only if R is an A -ring (resp., S A -ring). [ABSTRACT FROM AUTHOR]

Several authors have been interested in cotorsion theories. Among these theories we figure the pairs (Pn, Pn ⊥), where Pn designates the set of modules of projective dimension at most a given integer n > 1 over a ring R. In this paper, we shall focus on homological properties of the class P-1 ⊥ that we term the class of P1-injective modules. Numerous nice characterizations of rings as well as of their homological dimensions arise from this study. In particular, it is shown that a ring R is left hereditary if and only if any P1-injective module is injective and that R is left semi-hereditary if and only if any P1-injective module is FP-injective. Moreover, we prove that the global dimensions of R might be computed in terms of P1-injective modules, namely the formula for the global dimension and the weak global dimension turn out to be as follows wgl-dim(R) = sup{fdR(M) : M is a P1-injective left R-module} and l-gl-dim(R) = sup{pdR(M) : M is a P1-injective left R-module}. We close the paper by proving that, given a Matlis domain R and an R-module M ∈ P1, HomR(M, N) is P1-injective for each P1-injective module N if and only if M is strongly flat. [ABSTRACT FROM AUTHOR]

Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring . Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field into the general setting of algebras over an arbitrary ring . For this sake, we introduce and study the notion of a fibered AF-ring over a ring . This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc. 19 (1979) 391-401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings consisting of two -algebras and such that , we introduce the inherent notion to of a -fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product . As an application, we provide a formula for the Krull dimension of when either or is zero-dimensional as well as for the Krull dimension of when is a fibered AF-ring over the ring of integers with nonzero characteristic and is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of when is a Boolean ring. Actually, we prove that if and are rings such that is not trivial and is a Boolean ring, then dim. [ABSTRACT FROM AUTHOR]

The theory of Gorenstein flat dimension is not complete since it is not yet known whether the category 풢ℱ(R) of Gorenstein flat modules over a ring R is projectively resolving or not. Besides, it arises from recent investigations on this subject that there exists several ways of measuring the Gorenstein flat dimension of modules which turn out to coincide with the usual one in the case where 풢ℱ(R) is projectively resolving. These alternate procedures yield new invariants which enjoy very nice behavior for an arbitrary ring R. In this paper, we introduce and study one of these invariants called the cover Gorenstein flat dimension of a module M and denoted by R(M). This new entity stems from a sort of a Gorenstein flat precover of M. First, for each R-module M, we prove that R(M) ≤ R(M) for each R-module M with whenever R(M) is finite. Also, we show that 풢ℱ(R) is projectively resolving if and only if the Gorenstein flat dimension and the introduced cover Gorenstein flat dimension coincide. In particular, if R is a right coherent ring, then R(M) = R(M) for any R-module M. As a consequence, we prove that if R is a left and right GF-closed, then the Gorenstein weak global dimension of R is left-right symmetric and it is related to the cohomological invariants leftsfli(R) and rightsfli(R) by the formula [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give, via totally different techniques, an alternate proof to the main theorem of [18] in the category of modules over an arbitrary ring R. In effect, we prove that this theorem follows from establishing a sequence of equalities between specific classes of R-modules. Actually, we tackle the following natural question: What notion emerges when iterating the very process applied to build the Gorenstein projective and Gorenstein injective modules from complete resolutions? In other words, given an exact sequence of Gorenstein injective R-modules G= ⋯ → G1→ G0→ G-1→ ⋯ such that the complex R(H, G) is exact for each Gorenstein injective R-module H, is the module (G0→ G-1) Gorenstein injective? We settle such a question in the affirmative and the dual result for the Gorenstein projective modules follows easily via a similar treatment to that used in this paper. As an application, we provide the Gorenstein versions of the change of rings theorems for injective modules over an arbitrary ring. [ABSTRACT FROM AUTHOR]

This paper is concerned with the second change of rings theorems for the Gorenstein homological dimensions. More precisely, we give the ultimate versions of these theorems relatively to these new invariants. [ABSTRACT FROM AUTHOR]

Sather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module? [ABSTRACT FROM PUBLISHER]

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45-67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to explore new aspects of the prime ideal structure of tensor products of algebras over a field k. We prove that given a k-algebra A and a normal field extension K of k (in the sense of Galois theory), then for any prime ideals P1 and P2 of K ⊗k A lying over a fixed prime ideal p of A, there exists a k-automorphism σ of K such that (σ ⊗kidA)(P1) = P2. As an Application, we establish a result related to the dimension theory of tensor products stating that, for two arbitrary k-algebras A and B, the minimal prime ideals of p ⊗k B + Aσk q have the same height, for any prime ideals p and q of A and B, respectively. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the study of the dimension theory of tensor products of algebras over a field k. We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) and prove that any k-algebra A such that the polynomial ring in one variable A[X] is an AF-domain is in fact a GAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimension of A⊗kB for any k-algebra A such that A[X] is an AF-domain and any k-algebra B generalizing the main theorem of Wadsworth in [16]. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of the dimension theory of tensor products of algebras over a field k. We answer an open problem set in [6] and compute dim(A ⊗k B) when A is a k-algebra arising from a specific pullback construction involving AF-domains and B is an arbitrary k-algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, that such a pullback A is in fact a generalized AF-domain. [ABSTRACT FROM AUTHOR]

This article is concerned with the dimension theory of tensor products of algebras over a field k . In fact, we provide formulas for the Krull and valuative dimension of A ⊗k B when A and B are k -algebras such that the polynomial ring A [n] is an AF-domain for some positive integer n . Also, we compute dimv (A ⊗k B ) in the case where A ⊆ B . [ABSTRACT FROM AUTHOR]

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]