Concerning maximal $${\ell}$$ ℓ -ideals of rings of continuous integer-valued functions

Abstract

Maximal \({\ell}\)-ideals of the ring \({C(X, \mathbb{Z})}\) of continuous integer-valued functions on a topological space X were characterised by Subramanian to be exactly the minimal prime ideals of this ring. This note supplements this result by showing that, in fact, these ideals are also exactly the maximal z-ideals, exactly the maximal d-ideals, and exactly the maximal pure ideals of this ring. Casting everything in the category ODFrm of zero-dimensional frames, we show that what has just been said holds in any ring \({\mathfrak{Z}L}\) of continuous integer-valued functions on a zero-dimensional frame L. The ideals in question are describable in terms of the cozero map. They are precisely the inverse images (under the cozero map) of the points of the universal zero-dimensional compactification of L.