The ring of α-quotients

Abstract

This paper introduces, for each regular, uncountable cardinal α, the ring of quotients of the commutative ring A obtained by the direct limit QαA = lim→ HomA(I,A), where I ranges over the filter of base of ideals which can be generated by fewer than o elements of A. This is the ring of α-quotients. It is shown that QαA is the least α-selfinjective ring of quotients of A; that is to say, having the injective property relative to maps out of ideals generated by fewer than o elements. For semiprime rings, the ring of α-quotients of A has the α-splitting property: if D and D' are two subsets of size < α, such that dd' = 0, for each d ∈ D and d' ∈ D', and D U D' generates a dense ideal, then there is an idempotent e such that de = d, for all d ∈ D and de = 0, for all d ∈ D'. The paper examines the least ring of quotients QαsA with the α-splitting property. The application of greatest interest here is to archimedean f-rings. A considerable amount of attention is paid to the maximal l-ideal spaces of the rings QαA and QαsA. It is shown that the minimum α-cloz cover of mA, the space of maximal l-ideals of A, is none other than mQαsA. Applied to C(X), the ring of continuous real valued functions on a compact Hausdorff space X, it turns out that the minimum α-cloz cover of X is, in fact, mQαC(X), as long as X is zero-dimensional. Moreover, it is shown that A has the α-splitting property if and only if mA is α-cloz. The final section is devoted to the cardinal w1, and to the question of whether the minimum quasi F-cover of mA is mQw1 A. This is shown to be so, provided that A is complemented or mA is zero-dimensional.