Strongly $FP$-injective dimensions and Gorenstein projective precovers

TL;DR

This paper investigates the existence of Gorenstein projective precovers over rings, addresses open questions in the area, and confirms the Gorenstein Symmetry Conjecture under specific conditions.

Contribution

It provides new results on Gorenstein projective precovers, the completeness of related cotorsion pairs, and verifies the Gorenstein Symmetry Conjecture in certain cases.

Findings

Confirmed the existence of Gorenstein projective precovers over rings.

Proved the completeness of the Gorenstein projective cotorsion pair.

Validated the Gorenstein Symmetry Conjecture under specific conditions.

Abstract

The existence of the Gorenstein projective precovers over $R$ an arbitrary ring, as well as the completeness of the Gorenstein projective cotorsion pair $(\mathcal{GP},\mathcal{GP}^{\perp})$, are open questions. In this paper, we provide some answers to these questions and use the tool developed to confirm the Gorenstein Symmetry Conjecture, under two different situations. We also analyze situations where the finiteness of $\mathrm{silp}(R)$ implies $\mathrm{spli}(R)$ finite, and its counterpart.