Distributive extensions of modules

Let X be a submodule of a module M. The extension is said to be distributive if X ∩ (Y + Z) = X ∩ Y + X ∩ 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) is a distributive extension; (2) for any submodule Y of the module M, no simple subfactor of the module X/(X∩Y ) is isomorphic to any simple subfactor of Y/(X∩Y) (3) for any two elements x ∈ X and m ∈ M, there does not exist a simple factor module of the cyclic module xA/(X ∩ mA) that is isomorphic to a simple factor module of the cyclic module mA/(X ∩ mA). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 141–150, 2006.