Another ring-theoretic characterization of boundary spaces (Q2822587)

A Tychonoff space \(X\) is a \textit{boundary space} (or \textit{\(\partial\)-space}) if the boundary of every zero-set is contained in a zero-set with empty interior. Boundary spaces are precisely those spaces \(X\) for which every prime ideal of \(C(X)\) that consists entirely of zero-divisors is a \(d\)-ideal. In the present paper the authors extend this notion into the category of frames and frame homomorphisms and use then the language and techniques of pointfree topology to obtain the following new ring-theoretic characterization of boundary spaces: ``A Tychonoff space \(X\) is a boundary space if and only if \(C(X)\) is a boundary ring'', where a ring \(A\) is a \textit{boundary ring} if, for every \(a \in A\), the ideal \(M(a) + \text{Ann}(a)\) contains a non-divisor of zero (here \(M(a)\) denotes the intersection of all maximal ideals of \(A\) containing \(a\), while \(\text{Ann}(a)\) designates the annihilator of \(a\)).NEWLINENEWLINEAmong other results, they also show that a completely regular frame \(L\) is a boundary frame if and only if the Lindelöf coreflection of \(L\) is a boundary frame, if and only if the realcompact coreflection of \(L\) is a boundary frame.