Communications of the
Korean Mathematical Society CKMS

Let $R$ be a commutative ring with identity, and $\phi : \mathscr{I}(R) \rightarrow \mathscr{I}(R) \cup \{\varnothing\}$ be a function where $\mathscr{I}(R)$ is the set of all ideals of $R$. Following \cite{ab}, 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 $\operatorname{Spec}_{\phi}(R)$, has a Zariski topology analogous to that of the prime spectrum $\operatorname{Spec}(R)$, and show that this topological space is Noetherian if and only if $\phi$-radical ideals of $R$ satisfy the ascending chain condition.

Keywords: $\phi$-prime ideal, $\phi$-radical of an ideal, $\phi$-prime spectrum, $\phi$-top ring

MSC numbers: 13A15, 13A99