The ring of $ α $ -quotients

Abstract

This paper introduces, for each regular, uncountable cardinal \( \alpha \), the ring of quotients of the commutative ring A obtained by the direct limit ¶¶$ Q_\alpha A = \def\limind{\mathop{\oalign{\hfil$\rm lim$\hfil\cr$\longrightarrow$\cr}}} {\limind_} {\hom_A} {(I,A)} $¶¶ where I ranges over the filter of base of ideals which can be generated by fewer than \( \alpha \) elements of A. This is the ring of $ \alpha $ -quotients. It is shown that \( Q_\alpha A \) is the least \( \alpha \)-selfinjected ring of quotients of A; that is to say, having the injective property relative to maps out of ideals generated by fewer than a elements. For semiprime rings, the ring of \( \alpha \)-quotients of A has the $ \alpha $ -splitting property: if D and D' are two subsets of size \( $<$ \alpha \), such that dd' = 0, for each \( d \in D \) and \( d' \in D' \), and \( D \bigcup D' \) generates a dense ideal, then there is an idempotent e such that de = d, for all \( d \in D \) and de = 0, for all \( d \in D' \). The paper examines the least ring of quotients \( Q{^S}{_\alpha}A \) with the \( \alpha \)-splitting property. The application of greatest interest here is to archimedean f-rings. A considerable amount of attention is paid to the maximal \( \ell \)-ideal spaces of the rings \( Q_\alpha A \) and \( Q{^S}{_\alpha} A \). It is shown that the minimum \( \alpha \)-cloz cover of mA, the space of maximal \( \ell \)-ideals of A, is none other than \( \textrm{m}Q{^S}{_\alpha}A \). Applied to C(X), the ring of continuous real valued functions on a compact Hausdorff space X, it turns out that the minimum \( \alpha \)-cloz cover of X is, in fact, \( \textrm{m}Q{_\alpha}C(X) \), as long as X is zero-dimensional. Moreover, it is shown that A has the \( \alpha \)-splitting property if and only if mA is \( \alpha \)-cloz. The final section is devoted to the cardinal \( w_1 \), and to the question of whether the minimum quasi F-cover of mA is \( \textrm{m}Q_{w1}A \). This is shown to be so, provided that A is complemented or mA is zero-dimensional.