Commutative rings I

Let R be a commutative ring. An R-module M is said to be w-flat if Tor1 (M,N) is GV -torsion for any R-module N . It is known that every flat module is w-flat, but the converse is not true in general. The w-flat dimension of a module is defined in terms of w-flat resolutions. In this paper, we study the w-flat dimension of an injective w-module. To do so, we introduce and study the so-called w-copure (resp., strongly w-copure) flat modules and the w-copure flat dimensions for modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. We also study change of rings theorems for the w-copure flat dimension in various contexts. Finally some illustrative examples regarding the introduced concepts are given.

On $w$-copure flat modules and dimension

On strongly Gorenstein hereditary rings

For an integer N ≥ 2 and three subcategories &#x1d4b2;,&#x1d4b3;,&#x1d4b4; of modules, we study the notion of (&#x1d4b2;,&#x1d4b3;,&#x1d4b4;)-Gorenstein N-complexes. We show that under certain hypotheses, an N-complex C is (&#x1d4b2;,&#x1d4b3;,&#x1d4b4;)-Gorenstein if ...

Gorenstein objects in the category of N-complexes

Let R be a commutative ring with identity. An R-module E is said to be weak injective if ExtR1(N,E)=0 for any super finitely presented R-module N. In this article, for a domain R with quotient fiel...

Generalized coherent domains of self-weak injective dimension at most one

. In this paper, we mainly discuss Gorenstein Dedekind do-mains (G-Dedekind domains for short) and their overrings. Let R be aone-dimensional Noetherian domain with quotient ﬁeld K and integralclosure T. Then it is proved that R is a G-Dedekind domain if and onlyif for any prime ideal P of R which contains (R : K T), P is Gorensteinprojective. We also give not only an example to show that G-Dedekinddomains are not necessarily Noetherian Warﬁeld domains, but also a deﬁ-nition for a special kind of domain: a 2-DVR. As an application, we provethat a Noetherian domain R is a Warﬁeld domain if and only if for anymaximal ideal M of R, R M is a 2-DVR. IntroductionThroughout this paper, all rings are commutative with identity element andall modules are unitary. Let Rbe a domain with quotient ﬁeld K and J afractional ideal of R. The deﬁnition of J −1 can be found in [14] as follows:J −1 = {x| x∈ K, xJ⊂ R}.The deﬁnition of divisorial ideals can be found in [10]: A fractional ideal JofRis said to be divisorial if J

On overrings of gorenstein dedekind domains

In this note, we mainly discuss the Gorenstein Prüfer domains. It is shown that a domain is a Gorenstein Prüfer domain if and only if every finitely generated ideal is Gorenstein projective. It is also shown that a domain is a PID (resp., Dedekind domain, Bézout domain) if and only if it is a Gorenstein Prüfer UFD (resp., Krull domain, GCD domain).

A note on gorenstein prüfer domains

Let [Formula: see text] be a one-dimensional Noetherian domain and [Formula: see text] be a two-generated fractional ideal of [Formula: see text]. In this paper, we prove that [Formula: see text] is [Formula: see text]-projective if and only if [Formula: see text] and [Formula: see text] are equivalent, i.e. there exists a projective fractional ideal [Formula: see text] of [Formula: see text] such that [Formula: see text]. We also give an example to show that [Formula: see text] being [Formula: see text]-projective does not necessarily imply that [Formula: see text] and [Formula: see text] are isomorphic.

A note on two-generated ideals over domains

We study some homological properties of Gorenstein [Formula: see text]-injective modules, and we prove (1) a ring [Formula: see text]is not necessarily coherent if every Gorenstein [Formula: see text]-injective [Formula: see text]-module is injective, and (2) a ring [Formula: see text] is not necessarily coherent if every Gorenstein injective [Formula: see text]-module is injective. In addition, we characterize [Formula: see text]-Noetherian rings in terms of Gorenstein [Formula: see text]-injective modules, and we prove that a ring [Formula: see text] is [Formula: see text]-Noetherian if and only if every GV-torsion-free FP-injective [Formula: see text]-module is Gorenstein [Formula: see text]-injective, if and only if any direct sum of GV-torsion-free FP-injective [Formula: see text]-modules is Gorenstein [Formula: see text]-injective.

Gorenstein FP∞-Injective Modules and w-Noetherian Rings

Abstract In this paper, we show that every SG-projective ideal of a one dimensional Noetherian domain is 2-generated. We also prove that an integral domain is an NWF domain if and only if it is weakly SG-hereditary.

Kui Hu

Papers

On overrings of gorenstein dedekind domains

A note on gorenstein prüfer domains

A note on two-generated ideals over domains

Gorenstein FP∞-Injective Modules and w-Noetherian Rings

SG-projective ideals in one dimensional Noetherian domains