Let $ R $ be an arbitrary ring and $ T $ be a submodule of an $ R $-module $ M $. A
submodule $ N $ of $ M $ is called $ T $-small in $ M $ provided for each submodule $ X …

In this paper, we extend the concepts of a fuzzy essential submodule and a fuzzy
complement submodule to the concepts of fuzzy essential submodule with respect to an …

We introduce and study the concept of D-essential submodules and its dual concept, F-small
submodules, where D and F are arbitrary submodules of a module M. These two notions are …

S-系的D-多余子系 Page 1 山东大学学报(理学版)2023年11月第58卷第11期 E-mail:xblxb@sdu.edu.cn
Journal of Shandong University(Natural Science)ꎬ Vol.58ꎬ No.11ꎬ 2023 http:/ / lxbwk.njournal.sdu.edu.cn …

Through this paper R represent a commutative ring with identity and all R-modules are
unitary left R-modules. In this work we consider a generalization of the class of essential …

In this paper we Proved other properties of essential and complement submodules to an
arbitrary submodule of an R-module M. We prove that for a family {Mα} α∊ ʌ of modules. If …

‎ In this paper‎,‎ our aim is to introduce and study the essential submodules of an $ R $-
module $ M $ relative to an arbitrary submodule $ T $ of $ M $‎.‎ Let $ T $ be an arbitrary …

Let R be an associative ring with non-zero identity and M be a left R-module. A submodule N
of M is called annihilator small (briefly a-small), if for every submodule L of M with N+ L= M …

In this paper, T is a commutative ring with identity. We were interested in providing A new
conclusion about 𝛿-small submodule by clarifying the connection between the lifting module …

In this paper, we extend the concepts of a fuzzy essential submodule and a fuzzy
complement submodule to the concepts of fuzzy essential submodule with respect to an …