A TOPOLOGICAL APPROACH TO MORITA EQUIVALENCE FOR RINGS WITH LOCAL UNITS

In [1] and [3] a theory of Morita equivalence has recently been developed for certain not necessarily unital rings called rings with local units. In this article we prove that the special Horn-sets which figure in the description of equivalence functors are actually the sets of continuous homomorphisms from a locally projective generator (endowed with a suitable topology) into discrete modules. The main result of this paper says that two rings with local units which fulfill a topological condition of projectivity are Morita equivalent if and only if suitable matrix rings over them are isomorphic to each other.