\(\mathcal{P}_1\)-covers over commutative rings (Q2039343)

Let \(R\) be a ring, \(M\) a left \(R\)-module, and \(\mathcal{C}\) a class of left \(R\)-modules. A \(\mathcal{C}\)-precover of \(M\) is an \(R\)-homomorphism \(\phi: C \rightarrow M\) where \(C\in \mathcal{C}\) with the property that for any \(R\)-homomorphism \(\phi': C' \rightarrow M\) with \(C'\in \mathcal{C}\), there is an \(R\)-homomorphism \(f: C' \rightarrow C\) such that \(\phi'=\phi f\). A \(\mathcal{C}\)-precover \(\phi: C \rightarrow M\) is said to be a \(\mathcal{C}\)-cover if any \(R\)-homomorphism \(f: C \rightarrow C\) with \(\phi=\phi f\) is an isomorphism. The class \(\mathcal{C}\) is said to be covering if every left \(R\)-module admits a \(\mathcal{C}\)-cover. A famous theorem of Enochs says that if \(\mathcal{C}\) is closed under direct limits, then any left \(R\)-module that has a \(\mathcal{C}\)-precover has also a \(\mathcal{C}\)-cover. The converse problem, that is, if \(\mathcal{C}\) is covering, then \(\mathcal{C}\) is closed under direct limits, is known as Enochs' conjecture. In this paper, the authors consider the class \(\mathcal{P}_{1}(R)\) of \(R\)-modules of projective dimension at most one over a commutative ring \(R\), and investigate Enochs' conjecture for this class. They provide an affirmative answer in the case of a commutative semi-hereditary ring \(R\). Moreover, they describe the class \(\varinjlim \mathcal{P}_{1}(R)\) over (not necessarily commutative) rings which admit a classical ring of quotients.