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

Abstract

In this paper we consider the class P1(R) of modules of projective dimension at most one over a commutative ring R and we investigate when P1(R) is a covering class. More precisely, we investigate Enochs' Conjecture for this class, that is the question of whether P1(R) is covering necessarily implies that P1(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 (P1(R), P1(R) ⊥ ) which is not necessarily of finite type such that P1(R) satisfies Enochs' Conjecture. Moreover, we describe the class lim − → P1(R) over (not-necessarily commutative) rings which admit a classical ring of quotients.