Abstract
In this paper, we study a new concept of fuzzy sub-module, called fuzzy socle semi-prime sub-module that is a generalization the concept of semi-prime fuzzy sub-module and fuzzy of approximately semi-prime sub-module in the ordinary sense. This leads us to introduce level property which studies the relation between the ordinary and fuzzy sense of approximately semi-prime sub-module. Also, some of its characteristics and notions such as the intersection, image and external direct sum of fuzzy socle semi-prime sub-modules are introduced. Furthermore, the relation between the fuzzy socle semi-prime sub-module and other types of fuzzy sub-module presented.
Highlights
The concept of fuzzy sets was introduced by Zadeh in1965[1]
F-Soc-semi-prime sub-modules we offer the concept of an F-Soc-semi-prime sub-module as a generalization of ordinary concept(approximately semi-prime sub-module)
We were able to know some of the fuzzy algebraic properties of fuzzy socle semi-prime sub-modules and the relationship with other concepts
Summary
The concept of fuzzy sets was introduced by Zadeh in1965[1]. Many authors presented fuzzy subrings and fuzzy ideals. Definition 1.26 [3] A proper F-sub-module U of an F-module X of an R-module M is called semi-prime Fsub-module of X if whenever rbnmt ⊆ U,where rb is an F-singleton of R , mt is an Fsingleton of X and n ∈ Z+implies that rbmt ⊆ U for each t, b ∈ [0,1]. Let rb be an F-singleton of R and mt is an F-singleton of X , a proper F-sub-module U of an F-module X of an R-module M is called an F-Socle semi-prime ( for short F-Socsemi-prime) sub-module(ideal) of X if whenever rbnmt ⊆ U with n ∈ Z+ implies that rbmt ⊆ U + F − Soc(X) for each t, b ∈ [0,1].
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