Prime fuzzy ideals over noncommutative rings

TL;DR

This paper introduces the concept of prime fuzzy ideals in noncommutative rings, establishing their properties, equivalence with level cuts, and extending fuzzy ideal theory to noncommutative algebra.

Contribution

It defines prime fuzzy ideals in noncommutative rings, proves their equivalence with level cuts being crisp prime ideals, and develops foundational aspects of fuzzy noncommutative ring theory.

Findings

Prime fuzzy ideals are equivalent to level cuts being crisp prime ideals.

Semiprime fuzzy ideals are characterized as intersections of prime fuzzy ideals.

The paper introduces the fuzzy prime radical and foundational concepts for fuzzy noncommutative ring theory.

Abstract

In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in [Not-so-fuzzy fuzzy ideals, Fuzzy Sets and Systems 37 (1990), 237--243], which lacks this equivalence in a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes. This allows us to introduce the fuzzy prime radical and contribute to establish the basis of a Fuzzy Noncommutative Ring Theory.