LAUSR

Abstract The Cohn localization is a process of adjoining universal inverse morphisms to the category of R-modules over a noncommutative ring R. Finding specific models for universal localizations is, in general, difficult. For certain classes of rings, however, there exist constructions that are more accessible. We define such a class of rings, the generalized triangular matrix rings, and provide complete descriptions of their universal localizations with respect to certain classes of morphisms, generalizing results of Schofield and Sheiham. These descriptions are themselves generalized matrix rings. The results demonstrate, in particular, that localizing triangular matrix rings produces matrix rings with an increased degree of symmetry. This explains, in part, why Cohn localizations of triangular 2 × 2 matrix rings are always full matrix rings. However, the universal localization of a triangular matrix ring of order greater than two is often not fully symmetric and thereby not a full matrix ring. One consequence of this incomplete symmetry is the lack of a full set of matrix units to identify the localization. In spite of this difficulty, the localization does contain enough idempotents so that its entry bimodules can be recognized. In the case where the set of morphisms forms a maximal tree, the universal localization of any triangular matrix ring is a full matrix ring.