When certain prime ideals in rings of continuous functions are minimal or maximal

A Tychonoff space X is called a quasi m-space if every prime d  -ideal of the ring C(X)C(X) is either a maximal ideal or a minimal prime ideal. These spaces were characterised by Azarpanah and Karavan. In this paper we look at some properties of these spaces from a ring-theoretic perspective. We observe, for instance, that among subspaces which inherit this property are (i) cozero subspaces, (ii) dense z-embedded subspaces, and (iii) regular-closed subspaces among the normal quasi m-spaces. The ring-theoretic approach actually yields the above results within the broader context of frames. The latter part of the paper discusses completely regular frames L for which every prime z  -ideal in the ring RLRL is a maximal ideal or a minimal prime ideal.