Some Properties of Valuation Ideals and Primary Ideals in Additive Semigroups

M. Yahya Abbasi, Abul Basar

Abstract

This paper deals with the connections and interdependencies among prime ideals, primary ideals, valuation ideals, valuation semigroups and semigroups. R. Gilmer and J. Ohm [1] studied primary ideals and valuation ideals for integral domains. In this article,  we generalize this concept for semigroups. It is proved that if $T$ is a prime ideal of a semigroup $S$, and $\{Q_{\alpha }\}$ is the set of primary ideals that belongs to $T$. Further, if  $I=\bigcap Q_{\alpha}$ and every $Q_{\alpha}$ is a valuation ideal, then $I$ is prime.