Gorenstein Syzygy Modules

TL;DR

This paper characterizes Gorenstein n-syzygy modules as equivalent to n-syzygy modules and introduces the Gorenstein transpose concept, providing new insights into module theory over Noetherian rings.

Contribution

It establishes the equivalence of Gorenstein n-syzygy modules and n-syzygy modules, and introduces the Gorenstein transpose for finitely generated modules over Noetherian rings.

Findings

Gorenstein n-syzygy modules are equivalent to n-syzygy modules.

Introduces Gorenstein transpose for finitely generated modules.

Provides applications of Gorenstein transpose in module theory.

Abstract

For any ring $R$ and any positive integer $n$, we prove that a left $R$-module is a Gorenstein $n$-syzygy if and only if it is an $n$-syzygy. Over a left and right Noetherian ring, we introduce the notion of the Gorenstein transpose of finitely generated modules. We prove that a module $M\in \mod R^{op}$ is a Gorenstein transpose of a module $A\in \mod R$ if and only if $M$ can be embedded into a transpose of $A$ with the cokernel Gorenstein projective. Some applications of this result are given.