On (Strongly) Gorenstein (Semi)Hereditary Rings

Abstract

In this paper, we introduce and study the rings of Gorenstein homological dimensions less than or equal to 1. We call these Gorenstein (semi)hereditary rings and call a particular subclass of these strongly Gorenstein (semi)hereditary rings.

A Half-Centered Star-Operation on an Integral Domain

“…Recently, several classical results and notions on global homological dimensions have been extended to Gorenstein global homological dimensions. In the paper [29], the authors introduce the domains of Gorenstein homological dimensions at most one, which they call, by analogy to the classical ones, Gorenstein Dedekind and Prüfer domains, respectively. Although these domains come from the homological theory, they can also be characterized in terms of the ideal-theoretic concept.…”

Gorenstein semihereditary rings and Gorenstein Prüfer domains

“…The Gorenstein weak global dimension G-w.gl.dim(R) of R is defined as G-w.gl.dim(R) = sup{ Gfd R M | M is any R-module }. 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.…”

$\mathcal{GP}$-проективные и $\mathcal{GI}$-инъективные модули

“…Кольцо называется левым горенштейново наследственным [5], если всякий подмодуль проективного левого -модуля является горенштейновым проективным, т.е. l.Ggldim( ) 1.…”