All Gorenstein Hereditary Rings Are Coherent

Abstract

A ring R is called Gorenstein hereditary (G-hereditary) if every submodule of a projective module is Gorenstein projective (i.e. Ggldim (R) ≤ 1). In this paper, we settle a question raised by Mahdou and Tamekkante in [On (strongly) Gorenstein (semi)hereditary rings, Arab. J. Sci. Eng.36 (2011) 436] about the coherence of G-hereditary rings. It is shown that a ring R is Gorenstein semihereditary if and only if every finitely generated submodule of a projective module is Gorenstein projective. As a consequence of this result, we have that every G-hereditary ring R is coherent.

Gorenstein semihereditary rings and Gorenstein Prüfer domains

“…Recall that a ring R is called Gorenstein semihereditary [24] if it is a coherent ring with G-w.gl.dim(R) ≤ 1, ( i.e., R is a coherent ring such that all submodules of a flat R-module are Gorenstein flat). In [11], Gao and Wang shown that a ring R is Gorenstein semihereditary if and only if all finitely generated submodules of a projective R-module are Gorenstein projective. The Gorenstein semihereditary domains are called Gorenstein Prüfer domains in [28].…”

“…If R is a Gorenstein semihereditary ring, let M be a finitely generated submodule of a projective R-module P . By[11, Theorem 2.6], M is a finitely generated Gorenstein projective module. Since R is coherent, M is finitely presented.…”

Homological Properties of Modules with Finite Weak Injective and Weak Flat Dimensions

“…Recently, as a nice generalization of Stenström's viewpoint, Gao and Wang introduced the notions of weak injective and weak flat modules ( [11]). In this process, finitely presented modules were replaced by super finitely presented modules (see [10] or Sec. 2 for the definition).…”

Super Finitely Presented Modules and Gorenstein Projective Modules

“…In order to investigate the two questions above, we use the notion of so-called super finitely presented modules. Recall from [13] that an R-module M is said to be super finitely presented if it admits a projective resolution · · · → P n → · · · → P 1 → P 0 → M → 0 such that each P i is a finitely generated projective R-module. This notion has received attention in several papers in the literature.…”

“…In order to investigate the two questions above, we use the notion of so-called super finitely presented modules. Recall from [13] that an R-module M is said to be super finitely presented if it admits a projective resolution…”