Not-so-fuzzy fuzzy ideals

Prime fuzzy ideals over noncommutative rings

“…This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in [16], which lacks this equivalence in a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes.…”

On the Fuzzy Nil Radicals of Fuzzy Ideals of κ-semirings

“…By similar arguments of the Propositions 7-5, 7-6 and 7-7 of [7], we have the following propositions. …”

On the fuzzy polynomial ideals

“…Let α : R −→ [0, 1] be a fuzzy subset of a set R. We denote a level set α * by α * := {a ∈ R | α(a) = α(0)}. It was proved that if α is a fuzzy ideal of R, then α * is also a fuzzy ideal of R (Kumbhojkar and Bapat [2]). Note that if α is a fuzzy ideal of R, then α(0)…”

“…(Kumbhojkar and Bapat[2]) A fuzzy ideal α of a ring R is said to be a fuzzy prime ideal of R if α * is a prime ideal of R.Theorem 4.2. Let α : R −→ [0, 1] be a fuzzy ideal of a ring R. Then α is a fuzzy prime ideal of R if and only if α x is a fuzzy prime ideal of R[x].…”