$\mathcal P_1$-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, 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 $\varinjlim \mathcal P\_1(R)$ over (not necessarily commutative) rings which admit a classical ring of quotients.