ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS

We study primary ideals of the ring RL 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 RL is contained in a unique maximal ideal, and an ideal Q in RL is primary if and only if Q ∩ R∗L is a primary ideal in R∗L. We show that every pseudo-prime (primary) ideal in RL 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 RL is decomposable if and only if L = 2.