Generalizations of primal ideals over commutative semirings

Abstract

In this article we generalize some definitions and results fromideals in rings to ideals in semirings. Let $R$ be a commutativesemiring with identity. Let $\varphi \vartheta (R)\to \vartheta (R)\cup \{\mathrm{\varnothing }\}$ be a function, where $\vartheta (R)$ denotes the set of all ideals of $R$. A proper ideal $Iın\vartheta (R)$ is called $\varphi$-prime ideal if $raınI-\varphi (I)$implies $rınI$ or $aınI$. An element $aınR$ is called $\varphi$-prime to $I$ if $raınI-\varphi (I)$ (with $rınR$) implies that$rınI$. We denote by $p(I)$ the set of all elements of $R$ thatare not $\varphi$-prime to $I$. $I$ is called a $\varphi$-primal ideal of$R$ if the set $P=p(I)\cup \varphi (I)$ forms an ideal of $R$.Throughout this work, we define almost primal and $\varphi$-primalideals, and we also show that they enjoy many of the properties ofprimal ideals.