Generalizations of Prime Radical in Noncommutative Rings

Abstract

Let R be a noncommutative ring with identity. Let ϕ: S(R) → S(R) ∪ {∅} 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 Pϕ(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 Pϕ(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. © Palestine Polytechnic University-PPU 2024.