MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
Generalizations of primal ideals over commutative semirings
Malik Bataineh and Ruba Malas

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.

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Keywords: Primal ideal; $\phi$-prime ideal; weakly primal ideal; $\phi$-primal ideal.

MSC: 13A15, 16Y60

Pages:  133--139     

Volume  66 ,  Issue  2 ,  2014