Abstract
Abstract The cosets of a fuzzy ideal M in a ring A form another ring A/M, called the fuzzy-quotient ring of A and M. We show that A/M is always a fuzzy partition of A, and a fuzzy partition of A will be a fuzzy-quotient ring (with respect to some fuzzy ideal) if and only if its associated fuzzy similarity relation is compatible with the ring operations. We also discuss relationships between the two operations of the fuzzy-quotient ring A/M, and the extensions over IA of the two operations of A by Zadeh's extension principle. We study relationships between the level ring structures of a fuzzy-quotient ring and the level relations of its associated fuzzy similarity relations. We introduce and study a new notion of the factorization of a ring homomorphism through fuzzy-quotients. Finally, a relationship between fuzzy prime ideals and fuzzy similarity relation has been established.
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