On matrix rings over unit-regular rings (Q1866608)

The authors prove that a unit-regular ring \(R\) is isomorphic to a Busque ring \(S\), i.e., a directly finite countable-complete non-simple regular ring, if the matrix rings \(M_n(R)\) and \(M_n(S)\) are isomorphic for some positive integer \(n\). Note that the Busque rings are unit-regular. The result above answers positively an open problem of \textit{K. R. Goodearl}'s [in ``Von Neumann regular rings'' (1979; Zbl 0411.16007)], for the Busque rings.