Abstract
We investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study its properties. Given a fuzzy idealβofRand a homomorphismf:R→R′, we show that iffxis the induced homomorphism off, that is,fx(∑i=0naixi)=∑i=0nf(ai)xi, thenfx-1[(β)x]=(f-1(β))x.
Highlights
Zadeh [1] introduced the notion of a fuzzy subset A of a set X as a function from X into [0, 1]
We investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study its properties
Given a fuzzy ideal β of R and a homomorphism f : R → R, we show that if fx is the induced homomorphism of f, that is, fx(∑ni=0 aixi) = ∑ni=0 f(ai)xi, fx−1[(√β)x] = (√f−1(β))x
Summary
Zadeh [1] introduced the notion of a fuzzy subset A of a set X as a function from X into [0, 1]. Liu [3] introduced and studied the notion of the fuzzy ideals of a ring. The concept of fuzzy ideals was applied to several algebras: BN-algebras [5], BL-algebras [6], semirings [7], and semigroups [8]. The present authors [13] introduced the notion of a fuzzy polynomial ideal αx of a polynomial ring R[x] induced by a fuzzy ideal α of a ring R and obtained an isomorphism theorem of a ring of fuzzy cosets of αx. In this paper we investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study their properties
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