Fuzzy ideals in commutative rings

In this thesis, we are concerned with various aspects of fuzzy ideals of commutative rings. The central theorem is that of primary decomposition of a fuzzy ideal as an intersection of fuzzy primary ideals in a commutative Noetherian ring. We establish the existence and the two uniqueness theorems of primary decomposition of any fuzzy ideal with membership value 1 at the zero element. In proving this central result, we build up the necessary tools such as fuzzy primary ideals and the related concept of fuzzy maximal ideals, fuzzy prime ideals and fuzzy radicals. Another approach explores various characterizations of fuzzy ideals, namely, generation and level cuts of fuzzy ideals, relation between fuzzy ideals, congruences and quotient fuzzy rings. We also tie up several authors' seemingly different definitions of fuzzy prime, primary, semiprimary and fuzzy radicals available in the literature and show some of their equivalences and implications, providing counter-examples where certain implications fail

RESTRICTED CANCELLATION AND WEAKLY RESTRICTED CANCELLATION FUZZY MODULES

In this paper, we introduce and study (restricted cancellation, weakly restricted cancellation) fuzzy modules as generalization of notions restricted cancellation (weakly restricted cancellation) modules. We give many basic properties about both concepts

Fuzzy coprimary submodules

Let $R$ be a commutative ring with non-zero identity and let $M$ be a non-zero unitary $R$-module. The concept of fuzzy coprimary submodule as a dual notion of fuzzy primary one will be introduced and some of its properties will be studied. Among other things, the behavior of this concept with respect to fuzzy localization formation and fuzzy quotient will be examined