A test criterion for categorically compact rings

Abstract

The Kuratowski-Mrowka theorem - a topological space X is compact if and only if for each space Y the second projection map ∏₂:X x Y → Y is a closed map - provides for a categorical interpretation of compactness once a suitable notion of closure is obtained. In this article we study a general notion of compactness with respect to a class F of ring’s that is either hereditary semisimple or quotient reflective. We show that to test for categorical compactness of a ring R we need only check R R F, where R F = ∩{D D is an ideal of R and RD є F} , and to test ∏₂:(RRF)x H → H for rings H belonging to F as opposed to arbitrary H.