On prime and primary fuzzy ideals and their radicals

On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings

“…The following is due to Theorem 3.5 of [21]. (2) A fuzzy ideal which is prime according to Definition 5.1(i) (and hence according to Definition 5.1(iii)) is prime according to Definition 5.1(iv).…”

“…Let l be a fuzzy ideal of a commutative ring with identity R. Then According to [21], Definition 5.5(iv) may be considered as more suitable definition of primary fuzzy ideal. The following proposition is straightforward.…”

Strongly prime fuzzy ideals over noncommutative rings

“…Definition 10 ( [15]): Two fuzzy sets µ, ν of X are equivalent if for any x, y ∈ X, µ(x) > µ(y) ⇐⇒ ν(x) > ν(y). Definition 11 ([12]): A minimal prime ideal in a ring R is any prime ideal of R that does not properly contain any other prime ideals.…”

Radical Structures of Fuzzy Polynomial Ideals in a Ring

“…Proposition 16 (see [15] Proof. Let ( ) = 0 for all ∈ Ker and let ( ) = 0 + 1 + ⋅ ⋅ ⋅ + be any element of Ker .…”

“…A fuzzy ideal : → [0, 1] of a ring is called a fuzzy prime ideal [14] of if * is a prime ideal of . A fuzzy set √ : → [0, 1], defined as √ ( ) := ⋁{ ( ) | > 0}, is called a fuzzy nil radical [15] of . …”