Polyform and projective Σ-extending modules

By the results of [14, 15, 18], the following are equivalent for a ring R: (a) R is left non-singular and Sigma-extending; (b) R is a left and right hereditary serial artinian ring; (c) R is a left non-singular serial artinian QF-3-ring. Indeed, analogues of these remain equivalent if we replace R by a progenerator M in (a) and (c), and R by End(R)(M) in (b). Here, we study the interdependence of the statements for modules M which need not be progenerators or non-singular. We prove that a polyform Sigma-extending- module Pd is a direct sum of self-injective modules whose M-generated submodules are linearly ordered and satisfy ACC. Consequently, for M finitely generated, End(R)(M) is left and right serial, artinian hereditary. We show that a serial QF-3 module of finite length is Sigma-extending. Moreover, if M is finitely generated self-projective Sigma-extending with M-singular submodule Z(M)(M), then End(R)(M/Z(M)(M)) is left serial, artinian hereditary.