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 $\phi \:\vartheta (R)\rightarrow\vartheta (R)\cup \{\emptyset \}$ be a function, where $\vartheta(R)$ denotes the set of all ideals of $R$. A proper ideal $Iın\vartheta (R)$ is called $\phi$-prime ideal if $raın I-\phi(I)$implies $rın I$ or $aın I$. An element $aın R$ is called $\phi$-prime to $I$ if $raın I-\phi (I)$ (with $rın R$) implies that$rın I$. We denote by $p(I)$ the set of all elements of $R$ thatare not $\phi$-prime to $I$. $I$ is called a $\phi$-primal ideal of$R$ if the set $P=p(I)\cup \phi(I)$ forms an ideal of $R$.Throughout this work, we define almost primal and $\phi$-primalideals, and we also show that they enjoy many of the properties ofprimal ideals.