Multiplication modules over non-commutative rings

It is proved that each submodule of a multiplication module over a regular ring is a multiplicative module. If $A$ is a ring with commutative multiplication of right ideals, then each projective right ideal is a multiplicative module, and a finitely generated $A$-module $M$ is a multiplicative module if and only if all its localizations with respect to maximal right ideals of $A$ are cyclic modules over the corresponding localizations of $A$. In addition, several known results on multiplication modules over commutative rings are extended to modules over not necessarily commutative rings.