# $P_{1}$-covers over commutative rings

### Silvana Bazzoni

Università degli Studi di Padova, Italy### Giovanna Le Gros

Università degli Studi di Padova, Italy

## Abstract

In this paper we consider the class $P_{1}(R)$ of modules of projective dimension at most one over a commutative ring $R$ and we investigate when $P_{1}(R)$ is a covering class. More precisely, we investigate Enochs' Conjecture, that is the question of whether $P_{1}(R)$ is covering necessarily implies that $P_{1}(R)$ is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring $R$. This gives an example of a cotorsion pair $(P_{1}(R),P_{1}(R)_{⊥})$ which is not necessarily of finite type such that $P_{1}(R)$ satisfies Enochs' Conjecture. Moreover, we describe the class $lim P_{1}(R)$ over (not necessarily commutative) rings which admit a classical ring of quotients.

## Cite this article

Silvana Bazzoni, Giovanna Le Gros, $P_{1}$-covers over commutative rings. Rend. Sem. Mat. Univ. Padova 144 (2020), pp. 27–43

DOI 10.4171/RSMUP/54