Singular f-rings which are α-G.C.D. rings

This article introduces α-G.C.D. rings. Let α be a cardinal. A, a commutative ring with identity, is an α-G.C.D. ring if for each S ⊆ A of cardinality less than α there exists a g.c.d. for S, i.e., an element a ∈ A such that (s) ⩽ (a) for all s ∈ S and if b ∈ A also satisfies (s) ⩽ (b) for all s ∈ S then (a) ⩽ (b). In Section 2, it is shown that for a zero-dimensional space X, C(X,Z) is an ω+-G.C.D. ring if and only if X is a P-space. In Section 3, it is shown that the validity of the statement: for every zero-dimensional space X, C(X,Z) is an α-G.C.D. ring if and only if X is a Pα-space, is equivalent to the nonexistence of a measurable cardinal. An example is given of a zero-dimensional space X, and a cardinal α, for which C(X,Z) is an α-G.C.D. ring yet X is not a Pα-space.