${\mathcal{P}}_1$-covers over commutative rings

Abstract

In this paper we consider the class ${\mathcal{P}}_1(R)$ of modules of projective dimension at most one over a commutative ring $R$ and we investigate when ${\mathcal{P}}_1(R)$ is a covering class. More precisely, we investigate Enochs' Conjecture for this class, that is the question of whether ${\mathcal{P}}_1(R)$ is covering necessarily implies that ${\mathcal{P}}_1(R)$ 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 $({\mathcal{P}}_1(R),{\mathcal{P}}_1(R{)}^{\perp })$ which is not necessarily of finite type such that ${\mathcal{P}}_1(R)$ satisfies Enochs' Conjecture. Moreover, we describe the class $\underrightarrow{\lim }{\mathcal{P}}_1(R)$ over (not-necessarily commutative) rings which admit a classical ring of quotients.