Some Notes on z-Ideal and d-Ideal in $$\mathcal {R}L$$

Abstract

As usual, the ring of continuous real-valued functions on a completely regular frame L is denoted by $$\mathcal {R}L$$. It is well known that an ideal Q in $$\mathcal {R}L$$ is a z-ideal if and only if $$\sqrt{Q}$$ is a z-ideal in which case $$Q=\sqrt{Q}$$. We show the same fact in d-ideal context and then it turns out that the sum of a primary ideal and a z-ideal (d-ideal) in $$\mathcal {R}L$$ which are not in a chain is a prime z-ideal (d-ideal). For an ideal (a non-regular ideal) Q in $$\mathcal {R}L$$, we characterize the greatest z-ideal (d-ideal) contained in Q and the smallest z-ideal (d-ideal) containing Q in terms of basic z-ideals (d-ideals). We characterize frames L for which prime z-ideals and d-ideals coincide in $$\mathcal {R}L$$, or equivalently the sum of any two ideals consisting entirely of zero-divisors consists entirely of zero-divisors. Some characterizations of the quasi F-frame, extremally disconnected frames and P-frames in terms of d-ideals are provided. Finally, we show that a reduced ring R is regular if and only if every prime d-ideal in R is maximal.