# Gorenstein projective precovers and finitely presented modules

## TL;DR

This paper investigates conditions under which Gorenstein projective precovers exist over arbitrary rings, showing that focusing on finitely presented modules suffices to establish their existence in general.

## Contribution

It proves a reduction property indicating that verifying Gorenstein projective dimensions for finitely presented modules ensures the class of Gorenstein projective modules is special precovering.

## Key findings

- If all finitely presented modules have Gorenstein projective dimension ≤ n, then Gorenstein projective modules form a special precovering class.
- Over rings with finite Gorenstein global dimension, every module has a Gorenstein projective precover.
- The paper reduces the problem of existence of Gorenstein projective precovers to finitely presented modules.

## Abstract

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. It is known that if the ring has finite Gorenstein global dimension, then every module has a Gorenstein projective precover. We prove here a "reduction" property - we show that, over any ring, it suffices to consider finitely presented modules: if there exists a nonnegative integer $n$ such that every finitely presented module has Gorenstein projective dimension $\le n$, then the class of Gorenstein projective modules is special precovering.