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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]

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]

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]

Our main purpose in this paper is to characterize the AF-domain notion in terms of properties relative to chains of prime ideals in tensor products of algebras over a field k. Also, to illustrate our results, we investigate some characteristics of the spectrum of A ⊗k B in the case when the polynomial ring A [n] is an AF-domain for some positive integer n. [ABSTRACT FROM AUTHOR]