TY - JOUR ID - 1594 TI - ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS JO - Journal of Algebraic Systems JA - JAS LA - en SN - 2345-5128 AU - Abedi, M. AD - Esfarayen University of Technology, Esfarayen, North Khorasan, Iran. Y1 - 2020 PY - 2020 VL - 7 IS - 2 SP - 257 EP - 269 KW - Frame KW - primary ideal KW - pseudo-prime ideal KW - ring of continuous real-valued functions KW - decomposable ideal DO - 10.22044/jas.2019.8150.1399 N2 - 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 $L={\bf2}$. UR - https://jas.shahroodut.ac.ir/article_1594.html L1 - https://jas.shahroodut.ac.ir/article_1594_8108dcd6f42553890dfb9b0ba2055003.pdf ER -