On Modules and Matrix Rings With Sip-Extending

Abstract

In this note we study modules with the property that the intersection of two direct summands is essential in a direct summand (SIP-extending). Amongst other results we show that the class of right SIP-extending modules is neither closed under direct sums nor Morita invariant. Further we deal with direct summands of a SIP-extending module and SIP-extending matrix rings.

Modules close to SSP- and SIP-modules

“…essential in a direct summand of M for all i ∈ I, I is a finite index set, then i∈I A i is essential in a direct summand of M (see [16]). M is called a SSP-lifting module if A i lies above a direct summand of M for all i ∈ I, I is a finite index set, then i∈I A i lies above a direct summand of M (see [23]).…”

“…F. Karabacak and A. Tercan introduced the notion of SIP-CS module in [16]. 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 .…”

On Generalizations of Extending Modules

“…In [9], a module M is called an SIP-extending module provided that the intersection of every pair of direct summands of M is essential in a direct summand of M . We say a ring R is a right SIP-extending ring if the module R R is an SIP-extending module, i.e., for every pair of idempotents e, c in R there exists g 2 = g ∈ R such that eR ∩ cR is essential in gR.…”

“…In [9] we have provided a positive answer to the direct summand question under the condition that the summand is the unique closure of each of its essential submodules. Recall that R is said to Abelian if every idempotent of R is central.…”

Characterization of Some Well-Known Rings by Using Sa Modules

“…Günümüzde dik toplananlığa bağlı yeni tanımlar yapılarak bu anlamda birçok çalışmalar yapılmaktadır [1,4,7,8,10,11]. SIP özelliğine sahip modüller ilk olarak Wilson [16]…”