Distributive extensions of modules

Let $X$ be a submodule of a module $M$. The extension $X\subseteq M$ is said to be distributive if $X\cap(Y+Z)=X\cap Y+X\cap Z$ for any two submodules $Y$ and $Z$ of $M$. We study distributive extensions of modules over not necessarily commutative rings. In particular, it is proved that the following three conditions are equivalent: (1) $X_A\subseteq M_A$ is a distributive extension; (2) for any submodule $Y$ of the module $M$, no simple subfactor of the module $X/(X\cap Y)$ is isomorphic to any simple subfactor of $Y/(X\cap Y)$ (3) for any two elements $x\in X$ and $m\in M$, there does not exist a simple factor module of the cyclic module $xA/(X\cap mA)$ that is isomorphic to a simple factor module of the cyclic module $mA/(X\cap mA)$.