On a characterization of complete matrix rings
It was shown by P. R. Fuchs that a ring \(R\) is isomorphic to \(M_n(S)\) for some ring \(S\) and positive integer \(n\neq 1\) if and only if there are elements \(x\) and \(y\) of \(R\) such that (1) \(x^{n-1}\neq 0\), (2) \(x^n=0=y^2\), (3) \(x+y\) is a unit of \(R\), and (4) if \(r\) is an element of \(R\) with \(ryx^{n-1}=0\) then \(ry=0\). Fuchs also showed that (4) is redundant when \(n=2\). On the other hand, examples were constructed by Groenewald and the author to show that (4) does not follow from (1)-(3) if \(n\) is at least 3, but those examples were nevertheless \(n\) by \(n\) matrix rings. Here \(R\) is taken to be the direct sum of \(M_2(F)\) and \(M_n(F)\) for a field \(F\) and any positive integer \(n\) which is at least 3; there are elements \(x\) and \(y\) of \(R\) which satisfy conditions (1)-(3) but not (4); and it is easily shown that there is no positive integer \(k\neq 1\) and no ring \(S\) such that \(R\) is isomorphic to \(M_k(S)\).
scientific article; zbMATH DE number 938674
scientific article