Non-induced isomorphisms of matrix rings
If a ring \(R\) has the property that there are distinct positive integers \(i\) and \(j\) such that the free \(R\)-modules of ranks \(i\) and \(j\) are isomorphic, then the corresponding matrix rings \(M_i(R)\) and \(M_j(R)\) are isomorphic. The author shows that it is possible for \(M_i(R)\) and \(M_j(R)\) to be isomorphic even when the corresponding free modules are not isomorphic. In particular, a ring \(R\) is constructed with the properties that \(R\) is isomorphic to \(M_2(R)\) but \(R\) is not isomorphic as a module to the direct sum of two copies of itself. In a note added in proof, an outline is given of a subsequent construction due to Bergman which yields a ring \(R\) such that \(M_i(R)\) and \(M_j(R)\) are isomorphic for all positive integers \(i\) and \(j\), but the corresponding free modules are isomorphic if and only if \(i=j\).
scientific article; zbMATH DE number 1062239