Generalized "stacked bases" theorem for modules over semiperfect rings

The history of generalized "stacked bases" theorem origins from the result of Hill and Megibben on abelian groups. We extend this theorem for modules over semiperfect rings and as a consequence we show that for a submodule H of a projective module G over a semiperfect ring, the following conditions are equivalent: 1. there exists a decomposition G = circle plus P-i is an element of I(i) into a direct sum of indecomposable modules P-i, such that H = circle plus(i is an element of I)(P-i boolean AND H); 2. G/H is a direct sum of a family of modules, isomorphic to factor modules of principal indecomposable modules.