Modules in which the annihilator of a fully invariant submodule is pure

Abstract

A ring R is called left AIP if R modulo the left annihilator of any ideal is flat. In this paper, we characterize a module MR for which the endomorphism ring EndR(M) is left AIP. We say a module MR is endo-AIP (resp. endo-APP) if M has the property that “the left annihilator in EndR(M) of every fully invariant submodule of M (resp. EndR(M)m, for every m∈M) is pure as a left ideal in EndR(M)”. The notion of endo-AIP (resp. endo-APP) modules generalizes the notion of Rickart and p.q.-Baer modules to a much larger class of modules. It is shown that every direct summand of an endo-AIP (resp.endo-APP) module inherits the property and that every projective module over a left AIP (resp. APP)-ring is an endo-AIP (resp. endo-APP) module.