An extension of a factorization theorem of Wedderburn to matrix rings
Let \(R\) be a ring with identity, \(Z(R)\) the center of \(R\) and \(f(x)\in Z(R)[x]\) a monic polynomial of degree \(n\). Suppose \(f(a) = 0\) for some \(a \in R\). The authors show that under certain assumptions \(f(x) = (x-a_ 1)\cdots (x-a_ n)\). They develop an inductive formula to obtain the \(a_ i\) in \(R\). The procedure also works in a matrix situation and yields the following theorem. Let \(F\) be a field of characteristic zero and \(f(x) \in F[x]\) a monic polynomial of degree \(n\). Then there are nonderogatory \(n\times n\) matrices \(A_ 1,\dots, A_ n\) with entries in \(F\), all having characteristic polynomial \(f(x)\) such that \(f(x)I_ n = (xI_ n - A_ 1)\cdots (xI_ n - A_ n)\).
scientific article; zbMATH DE number 64491