Abstract

A ring R is said to have local units if every finite subset of it is contained in a subring with identity. If R is a ring with local units, then the family {R⋅(1−e)∣e∈R,e2=e} defines a ring topology τ on R. A ring R with local units is called topologically projective if the module (RR,τ) is a topological direct summand of a direct sum of discrete modules Re, where e is an idempotent from R. The main result of the paper states: Let R and S be topologically projective rings with local units. Then R and S are Morita equivalent if and only if there exists a set I for which the rings RI and SI are isomorphic, where RI and SI are the rings of I×I matrices over R and S respectively which contain at most finitely many nonzero entries.