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 M_R for which the endomorphism ring $En{d}_R(M)$ is left AIP. We say a module M_R is endo-AIP (resp. endo-APP) if M has the property that “the left annihilator in $En{d}_R(M)$ of every fully invariant submodule of M (resp. $En{d}_R(M)m,$ for every $m\in M$) is pure as a left ideal in $En{d}_R(M)$”. 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.

Keywords:

• Endo-AIP module
• endo-APP module
• left AIP ring
• left APP-ring
• pure ideal
• Rickart and p.q.-Baer modules
• s-unital ideal
• the endomorphism ring

1991 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D10
• 16D70