Generalizations of primal ideals in commutative rings

Abstract

Let $R$ be a commutative ring with identity. Let $\varphi :\text{sI}\to \text{eI}$be a function where $\text{sI}$ denotes the set of all ideals of $R$. Let$I$ be an ideal of $R$. An element $aınR$ is called $\varphi$-primeto $I$ if $raınI-\varphi (I)$ (with $rınR$) implies that $rınI$.We denote by $S_{\varphi }(I)$ the set of all elements of $R$ that are not$\varphi$-prime to $I$. $I$ is called a $\varphi$-primal ideal of $R$ ifthe set $P:=S_{\varphi }(I)\cup \varphi (I)$ forms an ideal of $R$. So if wetake ${\varphi }_{\mathrm{\varnothing }}(Q)=\mathrm{\varnothing }$ (resp., ${\varphi }_0(Q)=0$), a$\varphi$-primal ideal is primal (resp., weakly primal). In this paperwe study the properties of several generalizations of primal idealsof $R$.