  Let $R$ be a commutative ring. An $R$-module $M$ is said to be almost
projective if ${\rm Ext}^1_R(M, N) = 0$ for any $R_{\mathfrak{m}}$-module $N$
and any maximal ideal $\mathfrak{m}$ of $R$. In this paper, we investigate
rings $R$ over which every $R$-module has an almost projective envelope. We
also characterize rings $R$ over which every $R$-module has an almost
projective envelope with some special properties.
