The ring of \(\alpha \)-quotients.
The paper introduces, for each regular, uncountable cardinal \(\alpha \), the ring of quotients of the commutative ring \(\mathcal A\) obtained by the direct limit \[ {\mathcal Q}_{\alpha }\mathcal A=\varinjlim\Hom_{\mathcal A}(I,{\mathcal A}), \] where \(I\) ranges over the filter of base of ideals which can be generated by fewer than \(\alpha \) elements of \(\mathcal A\), the so-called ring of \(\alpha \)-quotients. It is shown that \(\mathcal Q_{\alpha }\mathcal A\) is the least \(\alpha \)-selfinjective ring of quotients of \(\mathcal A\). For semiprime rings, \(\mathcal Q_{\alpha }\mathcal A\) has the \(\alpha \)-splitting property. The results are applied namely to archimedean \(f\)-rings.
scientific article; zbMATH DE number 2158773