Generalizations of primal ideals in commutative rings

Ahmad Yousefian Darani

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 $a\in R$ is called $\phi$-prime to $I$ if $ra\in I - hi(I)$ (with $rn R$) implies that $rn I$. 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 $R$ if the set $P := S_\phi(I)\cup \phi(I)$ forms an ideal of $R$. So if we take $\phi_{\emptyset}(Q) = \emptyset$ (resp., $\phi_0(Q) = 0$), a $\phi$-primal ideal is primal (resp., weakly primal). In this paper we study the properties of several generalizations of primal ideals of $R$.