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.