%0 Journal Article
%T ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS
%J Journal of Algebraic Systems
%I Shahrood University of Technology
%Z 2345-5128
%A Abedi, M.
%D 2020
%\ 01/01/2020
%V 7
%N 2
%P 257-269
%! ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS
%K Frame
%K primary ideal
%K pseudo-prime ideal
%K ring of continuous real-valued functions
%K decomposable ideal
%R 10.22044/jas.2019.8150.1399
%X 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 $Q capmathcal{R}^*L$ 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 $L={bf2}$.
%U http://jas.shahroodut.ac.ir/article_1594_8108dcd6f42553890dfb9b0ba2055003.pdf