P_1 covers over commutative rings

In this paper we consider the class P_1(R) 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 for this class, 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)-orthogonal) which is not necessarily of finite type such that P_1(R) satisfies Enochs' Conjecture. Moreover, we describe the class P_1(R) over (not-necessarily commutative) rings which admit a classical ring of quotients.