ON QUASI-NORMALITY OF FUNCTION RINGS

An f-ring is called quasi-normal [26] if the sum of any two different minimal prime l-ideals is either a maximal l-ideal or the entire f-ring. Recall that the zero-component of a prime ideal P of a commutative ring A is the ideal Op = {a is an element of A vertical bar ab = 0 for some b is an element of A \ P}. Let C(X) be the f-ring of continuous real-valued functions on a Tychonoff space X. Larson proved that C(beta X) is quasinormal precisely when C(X) is quasinormal and the zero-component of every hyper-real ideal of C(X) is prime. We show that this result is actually purely ring-theoretic and thus deduce its extension to the f-rings RL of continuous real-valued functions on a frame L. A subspace of X is called a 2-boundary subspace if it is of the form cl(X) (C) boolean AND cl(X)(D) for some disjoint cozero-sets C and D of X. For normal spaces, Kimber [25] proved that C(X) is quasinormal precisely when every 2-boundary subspace of X is a P-space. By viewing spaces as locales, we obtain a characterization along similar lines which does not require normality, namely, for any Tychonoff space X, C(X) is quasi-normal if and only if every 2-boundary sublocale of the Lindelof reflection of X in the category of locales is a P-frame.