Abstract 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 $I\in
\vartheta (R)$ is called $\phi$prime ideal if $ra\in I\phi(I)$
implies $r\in I$ or $a\in I$. An element $a\in R$ is called $\phi
$prime to $I$ if $ra\in I\phi (I)$ (with $r\in R$) 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.
