Generalizations of primal ideals in commutative rings

Abstract

Let $R$ be a commutative ring with identity. Let $\phi: \sI\to \eI$be a function where $\sI$ denotes the set of all ideals of $R$. Let$I$ be an ideal of $R$. An element $aın R$ is called $\phi$-primeto $I$ if $raın I - \phi(I)$ (with $rın R$) implies that $rın I$.We denote by $S_\phi(I)$ the set of all elements of $R$ that are not$\phi$-prime to $I$. $I$ is called a $\phi$-primal ideal of $R$ ifthe set $P := S_\phi(I)\cup \phi(I)$ forms an ideal of $R$. So if wetake $\phi_{\emptyset}(Q) = \emptyset$ (resp., $\phi_0(Q) = 0$), a$\phi$-primal ideal is primal (resp., weakly primal). In this paperwe study the properties of several generalizations of primal idealsof $R$.