On Generalizations of Extending Modules

Abstract

A module M is said to be SIP-extending if the intersection of every pair of direct summands is essential in a direct summand of M. SIP-extending modules are a proper generalization of both SIP-modules and extending modules. Every direct summand of an SIP-module is an SIP-module just as a direct summand of an extending module is extending. While it is known that a direct sum of SIP-extending modules is not necessarily SIP-extending, the question about direct summands of an SIP-extending module to be SIP-extending remains open. In this study, we show that a direct summand of an SIP-extending module inherits this property under some conditions. Some related results are included about C11 and SIP-modules.

Strongly CS-Rickart and dual strongly CS-Rickart objects in abelian categories

“…Recall that an object M of an abelian category A has SIP (SSIP ) if for any family of (two) subobjects of M , their intersection is a direct summand, and it has SSP (SSSP ) if for any family of (two) subobjects of M , their sum is a direct summand (e.g., see [8]). Also, M is called SSIP-extending (SIP-extending) if for any family of (two) subobjects of M which are essential in direct summands of M , their intersection is essential in a direct summand of M , and SSSPlifting (SSP-lifting) if for any family of (two) subobjects of M which lie above direct summands of M , their sum lies above a direct summand of M [12] (also see [3,4,2,21] for modules). Next we consider some strict versions of SSIP-extending and SIP-extending objects, and their duals, SSSP-lifting and SSP-lifting objects in abelian categories.…”

CS-Rickart modules

“…A module M is called an SIP-CS module if the intersection of any two direct summands of M is essential in a direct summand of M. We say that a submodule N of a module M lies above a direct summand of M if there is a decomposition M = N 1 ⊕N 2 such that N 1 ⊂ N and N 2 ∩N small in N 2 . A module M is called an SSP-d-CS module if the sum of any two direct summands of M lies above a direct summand of M. While SIP-CS modules are proper generalizations of both SIP modules and CS modules (see [12,13]), SSP-d-CS are proper generalizations of both SSP modules and d-CS modules (see [22).…”