Document Type: Original Manuscript

Author

Esfarayen University of Technology, Esfarayen, North Khorasan, Iran.

Abstract

We study primary ideals of the ring $\mathcal{R}L$ of real-valued continuous functions on a completely regular frame $L$. We observe that prime ideals and primary ideals coincide in a $P$-frame. It is shown that every primary ideal in $\mathcal{R}L$ is contained in a unique maximal ideal, and an ideal $Q$ in $\mathcal{R}L$ is primary if and only if $Q \cap\mathcal{R}^*L$ is a primary ideal in $\mathcal{R}^*L$. We show that every pseudo-prime (primary) ideal in $\mathcal{R}L$ is either an essential ideal or a maximal ideal which is at the same time a minimal prime ideal. Finally, we prove that if $L$ is a connected frame, then the zero ideal in $\mathcal{R}L$ is decomposable if and only if $L={\bf2}$.

Keywords