A remark on commutative arithmetic rings

It is proved that a commutative ring with identity $R$ is arithmetic (i.e., the ideal lattice of $R$ is distributive) if and only if for any finitely generated (or any finitely presented) $R$-module $M$ and any ideal $I$ of $R$ the equality $I+\operatorname{Ann}M=\operatorname{Ann}(M/IM)$ holds.