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.

中文翻译：

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关于$$ \ mathcal {R} L $$ RL中的z -Ideal和d -Ideal的一些说明

通常，在完全规则的帧L上的连续实值函数的环由\（\ mathcal {R} L \）表示。众所周知，一个理想的Q在\（\ mathcal {R} L \）是一个Ž -理想当且仅当\（\ SQRT {Q} \）是一个ž -理想在这种情况下\（Q = \ sqrt {Q} \）。我们在d-理想情况下显示了相同的事实，然后证明不在链中的\（\ mathcal {R} L \）中的基本理想和z-理想（d -ideal）之和为素ž -理想（d-理想）。对于理想的（非规则理想）Q在\（\ mathcal {R} L \） ，我们表征最大Ž -理想（d -理想）包含在Q和最小Ž -理想（d含-理想）Q以基本z理想值（d理想值）表示。我们表征素L的z理想和d理想在\（\ mathcal {R} L \）中重合的框架，或者等效地，完全由零除数组成的两个理想之和全部由零除数组成。准F的一些特征-帧，提供了极端断开的帧和根据d-理想状态的P帧。最后，我们表明，当且仅当R中的每个素数d-理想值最大时，还原环R才是规则的。