Enochs' conjecture for small precovering classes of modules

Abstract: Enochs' conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this short paper, we prove the validity of the conjecture for small precovering classes, i.e. the classes of the form $\mathrm{Add}(M)$ where $M$ is any module, under a mild additional set-theoretic assumption which ensures that there are enough non-reflecting stationary sets. We even show that $M$ has a perfect decomposition if $\mathrm{Add}(M)$ is a covering class. Finally, the additional set-theoretic assumption is shown to be redundant if there exists an $n<\omega$ such that $M$ decomposes into a direct sum of $\aleph_n$-generated modules.