(n,m)-Strongly Gorenstein Projective Modules

TL;DR

This paper introduces and analyzes a new class of modules called $(n,m)$-strongly Gorenstein projective modules, exploring their properties and relationships with Gorenstein projective and flat dimensions.

Contribution

It defines the $(n,m)$-SG-projective modules, studies their syzygies, and establishes characterizations linking Gorenstein projective dimension with classical projective and flat dimensions.

Findings

Modules with Gorenstein projective dimension ≤ m are $(1,m)$-SG-projective when combined with a Gorenstein projective module.

Over rings with finite finitistic flat dimension, finite Gorenstein projective dimension implies finite projective dimension.

Abstract

This paper is a continuation of the papers J. Pure Appl. Algebra, 210 (2007), 437--445 and J. Algebra Appl., 8 (2009), 219--227. Namely, we introduce and study a doubly filtered set of classes of modules of finite Gorenstein projective dimension, which are called $(n,m)$-strongly Gorenstein projective ($(n,m)$-SG-projective for short) for integers $n\geq 1$ and $m\geq 0$. We are mainly interested in studying syzygies of these modules. As consequences, we show that a module $M$ has Gorenstein projective dimension at most $m$ if and only if $M\oplus G$ is $(1,m)$-SG-projective for some Gorenstein projective module $G$. And, over rings of finite left finitistic flat dimension, that a module of finite Gorenstein projective dimension has finite projective dimension if and only if it has finite flat dimension.