Abstract
Problem statement: Let M be a right module over a ring R. In this article modules in σ[M] with chain conditions on δM- small submodules are studied. Approach: With the help of known results about M- singular, Artinian and Noetherian modules the techniques of the proofs of our main results use the properties of δM- small, δM- supplement and δM- semimaximal submodules. Results: Modules in σ[M] with chain conditions on δM- small are investigated, δM- semimaximal submodule is defined . Some Properties ofδM- semimaximal submodules are proved. As application a new characterization of Artinian module in σ[M] is obtained in terms of δM- small submodules and δM- semimaximal submodules, as well as δM- small submodules and δM- supplement submodules. Conclusion/Recommendations: Our results certainly generalized several results obtained earlier.
Highlights
Throughout this research, R denotes an associative ring with unity and modules M are unitary right Rmodules Mod-R denotes the category of all right Rmodules
ΔM − small submodules are the generalization of δ- small submodules in the category Mod-R Let L,K be two submodules of M L is called a δ- supplement of Kin M if M= L+K and L ∩ K δ L
Recall that a module M is said to have a uniform dimension n, where n is a nonnegative integer,if n is the maximal number of summands in a direct sum of nonzero submodules of M
Summary
MATERIALS AND METHODSThroughout this research, R denotes an associative ring with unity and modules M are unitary right Rmodules Mod-R denotes the category of all right Rmodules. Zhou called a submodule N of a module M is δ- small in M ( notation N ≤δ M ) if, whenever N+X=M with If for every submodules L,K of M with M=L+K there exists a δ − supplement N of L in Msuch that N ≤ K, M is called an amply δ − supplemented module. On the other hand N is called an amply δM − supplemented module if for every submodules L,K with N= L+K there exists a δM − supplement X of L such that X ≤ K.
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