Abstract
This research analyses the correlation that exists between fuzzy T-ABSO and fuzzy socle. To answer this question, we define a new concept called, fuzzy socle T-ABSO submodules. Results showed many properties that satisfied the new notion. In addition, we applied some algebra operations and found the relationship between fuzzy T-ABSO and fuzzy socle -T-ABSO. We also investigated the structure of fuzzy socle T-ABSO sub- modules of a module under fuzzy direct sum, homomorphic image and intersection. Moreover, the relation between fuzzy socle T-ABSO sub-module and other types of fuzzy sub-module was presented. The implications of this study could be used to define new concepts that depend on fuzzy socle T-ABSO concepts
Highlights
In 1965, Zadeh [1] presented a fuzzy subset A of X as a function from X to the unit interval [0,1]
We investigated the structure of fuzzy socle T-ABSO sub- modules of a module under fuzzy direct sum, homomorphic image and intersection
The implications of this study could be used to define new concepts that depend on fuzzy socle T-ABSO concepts
Summary
In 1965, Zadeh [1] presented a fuzzy subset A of X as a function from X to the unit interval [0,1]. Rabi [11], in (2001), generalised this concept to a prime fuzzy submodule: Let X be a F-module of an R-module M, A F-submodule U of X is called prime if rbmt ⊆ U,with rbis F-singleton of R and mtis a F-singleton of X, implying that either mt ⊆ Uor rb ⊆ [U :R X] for each t, b ∈ [0, 1]’. Wafaa presented the concept of T-Absorbing fuzzy submodule: Let X be a F-module of an R-module M, A Fsubmodule U of X is called T-ABSO if sqrbmt ⊆ U,where sq, rbare F-singletons of R and mtis a F-singleton of X, implying that either rbmt ⊆ U or sqmt ⊆ Uor sqrb ⊆ [U :R X] for each t, q, b ∈ [0, 1]’[12]. Note: We denote fuzzy Socle T-Absorbing submodule by F-Soc-T-ABSO submodule
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