A GENERALIZATION OF THE PRIME RADICAL OF IDEALS IN COMMUTATIVE RINGS

A GENERALIZATION OF THE PRIME RADICAL OF IDEALS IN COMMUTATIVE RINGS

Let R be a commutative ring with identity, and ${\phi}:{\mathfrak{I}}(R){\rightarrow}{\mathfrak{I}}(R){\cup}\{{\varnothing}\}$ be a function where ${\mathfrak{I}}(R)$ is the set of all ideals of R. Following [2], a proper ideal P of R is called a ${\phi}$-prime ideal if $x,y{\in}R$ with $xy{\in}P-{\phi}(P)$ implies $x{\in}P$ or $y{\in}P$. For an ideal I of R, we define the ${\phi}$-radical ${\sqrt[{\phi}]{I}}$ to be the intersection of all ${\phi}$-prime ideals of R containing I, and show that this notion inherits most of the essential properties of the usual notion of radical of an ideal. We also investigate when the set of all ${\phi}$-prime ideals of R, denoted $Spec_{\phi}(R)$, has a Zariski topology analogous to that of the prime spectrum Spec(R), and show that this topological space is Noetherian if and only if ${\phi}$-radical ideals of R satisfy the ascending chain condition.