D-Extending Modules

Abstract

A submodule N of a module M is called d-closed if M/N has a zero socle. D-closed submodules are similar concept to s-closed submodules, which are defined through nonsingular modules by Goodearl. In this article we deal with modules with the property that all d-closed submodules are direct summands (respectively, closed, pure). The structure of a ring over which d-closed submodules of every module are direct summand (respectively, closed, pure) is studied.

C11 -modules via left exact preradicals

“…In other words, for every submodule N of M , r(M/N ) = 0 implies that N is a complement in M . For example, if r = Soc then r c -module and C -module definitions coincide (see [5]). Then, we have the following straightforward observation.…”

“…Extending modules and their generalizations play an important role in modules and rings. To this end, several generalizations of CS notion have been worked out extensively by many authors (see, for example [1,5,[8][9][10][12][13][14]). This kind of investigations are traced back to the theory of C 11 -modules as well as C 11 -rings.…”

“…It has come to our attention that the following general results in [13,Lemma 4,Theorem 9] has been missed out by the some other authors (see [1,5]). It seems that there is no direct way to achieve the latter paper.…”

“…We give the following easy example which shows that CS property does not imply the condition worked as d -extending in [5].…”

Addendum to "D-extending modules" [Hacet. J. Math. Stat. 49(3), 914-920, 2020]

“…

After the paper [1] was published, the following information about D-extendig modules was pointed out by Adnan Tercan.Let r be a left exact preradical in the category of right R-modules and let M be a rightcalled an ES-module provided every exact submodule of M is a direct summand of M . ES-modules was introduced by A. Tercan in [2].

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