Equiprime, 3-prime and c-prime fuzzy ideals of nearrings

Abstract

In this paper, we present the notions of equiprime fuzzy ideal, 3-prime fuzzy ideal and c-prime fuzzy ideal of a nearring. We characterize these fuzzy ideals using level subsets and fuzzy points. If f: N ? M is an onto nearring homomorphism, we show that the map l 7 ! f ðlÞ defines a one-to-one correspondence between the set of all f-invariant (alternatively with sup property) equiprime (3-prime and c-prime, respectively) fuzzy ideals of N and the set of all equiprime (3-prime and c-prime, respectively) fuzzy ideals of M. Finally, we define fuzzy cosets determined by generalized fuzzy ideals; obtain fundamental results and isomorphism theorems.

Interval valued L-fuzzy cosets of nearrings and isomorphism theorems

“…Let aμ ∈ N /μ. By Proposition 2.9 (ii) of Kedukodi, Jagadeesha and Kuncham [29] we get, C I (α,μ(a0)) ≥β ⇒ C I (α,μ(a0 − 0)) ≥β ⇒ a0μ = 0μ (by Theorem 4.4) ⇒ aμ · 0μ = 0μ ⇒ N /μ is a zerosymmetric nearring.…”

“…Let xμ , yμ ∈ N /μ such that xμ · yμ = 0μ ⇒ xyμ = 0μ ⇒ C I (α,μ(x y − 0))) ≥β (by Theorem 4.4). Asμ is an i-v c-prime ideal of N by Proposition 2.10 of Kedukodi et al [29] we get, C I (α,μ(x)) ≥β or C I (α,μ(y) ≥β ⇒ xμ = 0μ or yμ = 0μ .…”

“…Davvaz [15] introduced fuzzy R-subgroup of a nearring with thresholds and studied implication operators. Kedukodi et al [30,31] characterized equiprime, 3-prime and c-prime fuzzy ideals of nearring in terms of their level subsets. Bhavanari et al [11] defined the graph of a nearring with respect to an ideal.…”

Nearring ideals, graphs and cliques

“…For the following definitions and results we refer to Kedukodi, Kuncham and Bhavanari [12,13]. A fuzzy ideal µ of N is called equiprime if for all x, y, a ∈ N ,…”

“…Yuan, Zhang and Ren [18] gave the definition of a fuzzy subgroup with a lower threshold α and an upper threshold β. Davvaz [8] used this idea to define the concept of a fuzzy ideal with thresholds α and β. Kedukodi, Kuncham and Bhavanari [13] used the concept of thresholds to define the notions of equiprime fuzzy ideal, 3-prime fuzzy ideal and c-prime fuzzy ideal of a nearring N . The primary benefit of the concept of thresholds is the choice for thresholds which gives rise to the fuzzy character in the examples.…”