Generalizations of primal ideals over commutative semirings

Malik Bataineh, Ruba Malas

In this article we generalize some definitions and results from ideals in rings to ideals in semirings. Let $R$ be a commutative semiring 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 $Iı \vartheta (R)$ is called $hi$-prime ideal if $ra\in I-hi(I)$ implies $r\in I$ or $a\in I$. An element $a\in R$ is called $\phi $-prime to $I$ if $ra\in I-hi (I)$ (with $rı R$) implies that $r\in I$. We denote by $p(I)$ 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 $P=p(I)\cup \phi(I)$ forms an ideal of $R$. Throughout this work, we define almost primal and $\phi$-primal ideals, and we also show that they enjoy many of the properties of primal ideals.