INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REALVALUED CONTINUOUS FUNCTIONS ON A FRAME

A frame L is called coz-dense if Σcoz(α) = ∅ implies α = 0. Let RL be the ring of real-valued continuous functions on a coz-dense and completely regular frame L. We present a description of the socle of the ring RL, based on minimal ideals of RL and zero sets in pointfree topology. We show that socle of RL is an essential ideal in RL if and only if the set of isolated points of ΣL is dense in ΣL if and only if the intersection of any family of essential ideals is essential in RL. Besides, the counterpart of some results in the ring C(X) is studied for the ring RL. For example, an ideal E of RL is an essential ideal if and only if ∩ Z[E] is a nowhere dense subset of ΣL.