$\mathcal P_1$-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.