In this paper, we introduce and study the concept of S-coprime submodules, where a proper submodule N of an R-module M is called S-coprime submodule if M N is S-coprime Rmodule. Many properties about this concept are investigated.
Throughout this paper, we introduce the notion of weak essential F-submodules of F-modules as a generalization of weak essential submodules. Also we study the homomorphic image and inverse image of weak essential F-submodules.
The purpose of this paper is to introduce dual notions of two known concepts which are semi-essential submodules and semi-uniform modules. We call these concepts; cosemi-essential submodules and cosemi-uniform modules respectively. Also, we verify that these concepts form generalizations of two well-known classes; coessential submodules and couniform modules respectively. Some conditions are considered to obtain the equivalence between cosemi-uniform and couniform. Furthermore, the relationships of cosemi-uniform module with other related concepts are studied, and some conditional characterizations of cosemi-uniform modules are investigated.
Let R be a commutative ring with identity and E be a unitary left R – module .We introduce and study the concept Weak Pseudo – 2 – Absorbing submodules as generalization of weakle – 2 – Absorbing submodules , where a proper submodule A of an R – module E is called Weak Pseudo – 2 – Absorbing if 0 ≠rsx A for r, s R , x E , implies that rx A + soc ( E ) or sx A + soc (E) or rs [ A + soc ( E ) E ]. Many basic properties, char
... Show MoreIn this article, we introduce a class of modules that is analogous of generalized extending modules. First we define a module M to be a generalized ECS if and only if for each ec-closed submodule A of M, there exists a direct summand D of M such that is singular, and then we locate generalized ECS between the other extending generalizations. After that we present some of characterizations of generalized ECS condition. Finally, we show that the direct sum of a generalized ECS need not be generalized ECS and deal with decompositions for be generalized ECS concept.
Let R be an individual left R-module of the same type as W, with W being a ring containing one. W’s submodules N and K should be referred to as N and K, respectively that K ⊆ N ⊆ W if N/K <<_J (D_j (W)+K)/K, Then K is known as the D J-coessential submodule of Nin W as K⊆_ (Rce) N. Coessential submodule is a generalization of this idea. These submodules have certain interesting qualities, such that if a certain condition is met, the homomorphic image of D J- N has a coessential submodule called D J-coessential submodule.
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 submodules namely annihilator essential submodules. We study the relation between the submodule and his annihilator and we give some basic properties. Also we introduce the concept of annihilator uniform modules and annihilator maximal submodules.
Let be a commutative ring with unity and let be a submodule of anon zero left R-module , is called semiprime if whenever , implies . In this paper we say that is nearly semiprime, if whenever , implies ( ),(in short ),where ( )is the Jacobson radical of . We give many results of this type of submodules.
Let R be a ring and let A be a unitary left R-module. A proper submodule H of an R-module A is called 2-absorbing , if rsa∈H, where r,s∈R,a∈A, implies that either ra∈H or sa∈H or rs∈[H:A], and a proper submodule H of an R-module A is called quasi-prime , if rsa∈H, where r,s∈R,a∈A, implies that either ra∈H or sa∈H. This led us to introduce the concept pseudo quasi-2-absorbing submodule, as a generalization of both concepts above, where a proper submodule H of an R-module A is called a pseudo quasi-2-absorbing submodule of A, if whenever rsta∈H,where r,s,t∈R,a∈A, implies that either rsa∈H+soc(A) or sta∈H+soc(A) or rta∈H+soc(A), where soc(A) is socal of an
... Show MoreLet M be an R-module, where R is a commutative ring with unity. A submodule N of M is called e-small (denoted by N e  M) if N + K = M, where K e  M implies K = M. We give many properties related with this type of submodules.