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

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.