z∘-ideals in intermediate rings of ordered field valued continuous functions
Bag, Sagarmoy · Acharyya, Sudip Kumar · Mandal, Dhananjoy
Original · EN
A proper ideal I in a commutative ring with unity is called a z∘-ideal if for each a in I, the intersection of all minimal prime ideals in R which contain a is contained in I. For any totally ordered field F and a completely F-regular topological space X, let C(X,F) be the ring of all F-valued continuous functions on X and B(X,F) the aggregate of all those functions which are bounded over X. An explicit formula for all the z∘-ideals in A(X,F) in terms of ideals of closed sets in X is given. It turns out that an intermediate ring A(X,F)≠ C(X,F) is never regular in the sense of Von-Neumann. This property further characterizes C(X,F) amongst the intermediate rings within the class of PF-spaces X. It is also realized that X is an almost PF-space if and only if each maximal ideal in C(X,F) is z∘-ideal. Incidentally this property also characterizes C(X,F) amongst the intermediate rings within the family of almost PF-spaces.
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