On Semiprime and Quasi-Semiprime Ideals in Ordered AG-Groupoids

Abstract

In this paper, we investigate the notion of semiprime ideals in in ordered $\mathcal{AG}$-groupoids  as a generalization of prime ideals. The aim of this paper is to investigate the concept of semiprime and quasi-semiprime ideals in ordered $\mathcal{AG}$-groupoids with left identity. Moreover, we investigate relationships between semiprime and quasi-semiprime ideals in ordered $\mathcal{AG}$-groupoids. It is show that an ideal $\displaystyle\prod_{i\in I}P_{i}$ of an ordered $\mathcal{AG}$-groupoid $\displaystyle\prod_{i\in I}S_{i}$ is semiprime if and only if $\displaystyle\prod_{i\in I}(a_{i}(S_{i}a_{i})]\subseteq \displaystyle\prod_{i\in I}P_{i}$ implies that $(a_{i})_{i\in I}\in\displaystyle\prod_{i\in I}P_{i}$, where $(a_{i})_{i\in I}\in\displaystyle\prod_{i\in I}S_{i}$.