Enochs’ Conjecture for Small Precovering Classes of Modules

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