Centralizers in endomorphism rings. (Q627970)

Let \(R\) be a ring with centre \(Z(R)\) and let \(M\) be a finitely generated semisimple left \(R\)-module. Suppose \(\varphi\) is a nilpotent endomorphism of the module \(M\). The main result of the paper under review consists of a description of the centralizer \(C(\varphi)\) of \(\varphi\) in \(\text{End}_R(M)\). It turns out that \(C(\varphi)\) is a homomorphic image of the opposite of a \(Z(R)\)-subalgebra of the matrix algebra \(M_m(R[z])\), \(m=\dim\ker(\varphi)\). A precise description of \(C(\varphi)\) is obtained when \(R\) is a local ring. As an application, when \(K\) is a field, and \(A\in M_m(K)\), the PI degree of \(C(A)\) is determined as well.