GENERALIZATIONS OF PRIME RADICAL IN NONCOMMUTATIVE RINGS.

Let R be a noncommutative ring with identity. Let φ : S(R) -- S(R) → { Φ} be a function where S(R) denotes the set of all subsets of R. The aim of this paper is to generalize the concept of prime radical √I of an ideal I of R to φ-prime radical P_φ (I). A proper ideal Q of R is called φ-prime if whenever a, b ∈ R, aRb ⊆ Q and aRb ... φ (Q) implies that either a ∈ Q or b ∈ Q. In this paper, first we study the properties of several generalizations of prime ideals of R. Then, we verify that P _φ(I) is equal to the intersection of all minimal φ-prime ideals of R containing I, and we show that this notion inherits many of the essential properties of the usual notion of prime radical of an ideal. [ABSTRACT FROM AUTHOR]