Distributive extensions of modules

Let X be a submodule of a module M. The extension X M 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) XA MA 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). © 2008 Springer Science+Business Media, Inc.