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Duality of St-closed subodules and semi-extending modules
Abstract
In this article, we introduce the dual notions of St-closed submodules and semi-extending modules; we call them CSt-closed and cosemi-extending modules respectively. We investigate some basic properties of these classes of modules, and dualize others, which gave by Ahmed and Abbas. Moreover, we study the relationships of cosemi extending with other related concepts. For doing that we need to introduce some new classes such as Pr-supplemented and Pr-lifting modules.
... A Hadi and Ibrahiem introduced P-small submodules as an extension to the concept of small submodules, where a proper submodule N of an R-module M is called P-small (simply N M), if N+P≠M for every prime submodule P of M [6]. A generalization of coessential submodules appeared in another study [7], where a submodule L is called cosemi-essential of N in M, if . A submodule N is called coclosed in M (simply N M), if N has no proper coessential submodule in M [8]. ...
... Examples and Remarks (2.2) 1. Every CSt-Polyform module is copolyform, since every CSt-closed submodule is coclosed [7]; hence, the result follows directly from the definition of CSt-Polyform module. 2. The converse of (1) is not true in general; for example, the Z-module Z is copolyform. ...
... Note that Z/(4) Z 4 and (Z,Z 4 )=0. On the other hand, (0) Z 4 [7], thus Z is not CSt-Polyform. 3. Every simple module is CSt-Polyform module. ...
In the year 2018, the concept of St-Polyform modules was introduced and studied by Ahmed, where a module M is called St-Polyform, if for every submodule N of M and for any homomorphism :N M, ker is St-closed submodule in N. The novelty of this paper is that it dualizes this class of modules to a form that we denote as CSt-Polyform modules. Accordingly, some results that appeared in the original paper are dualized. For example, we prove that in the class of hollow modules, every CSt-Polyform module is Coquasi-Dedekind. In addition, several important properties of CSt-Polyform module are established, while further characterization of CSt-Polyform is provided. Moreover, many relationships of CSt-Polyform modules with other related concepts are considered, such as the copolyform, epiform, CSt-semisimple,-nonsingular modules, while some others will be introduced, such as the non-CSt-singular and G. Coquasi-Dedekind modules.
... Recall that a submodule U of an R-module Y is termed St-closed (briefly ≤ ), if U has no proper semi-essential extension in Y, i.e., if there exists submodule U of Y such that ≤ ≤ , then = , [16]. ...
Let be a commutative ring with 1 and be a left unitary R-module. In this paper, we give a generalization for the notions of compressible (retractable) module. As well as, we study closed (St-closed) compressible and closed (St-closed) retractable. Furthermore, some of their advantages, properties, categorizations and instances have been given. Finally, we study the relation between them.
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