Study of modules over formal triangular matrix rings

In this paper we carry out a systematic study of modules over a formal triangular matrix ring Using the alternative description of right T-modules as triples (X,Y)f with X Mod−A, Y Mod−B and f : Y B M → X in Mod−A, we shall characterize respectively uniform, hollow, finitely embedded, projective, generator or progenerator modules over T. For projective modules an explicit method for constructing a dual basis is described. Also necessary and sufficient conditions are found for a T-module to admit a projective cover. When the conditions are fulfilled we give an explicit method for constructing a projective cover.