Minimal polynomials of algebraic derivations and automorphisms.
Let \(R\) be a prime ring with extended centroid \(C\), and let \(a\in R\) be \(C\)-algebraic. The paper describes the minimal polynomials of the inner derivation \(x\mapsto[a,x]\) and (for an invertible \(a\)) the inner automorphism \(x\mapsto axa^{-1}\) in terms of the minimal polynomial of \(a\). The results are definitive of their kind, but somewhat too technical to be stated here. They are in fact derived from more general results on minimal polynomials of maps \(x\mapsto ax+xb\) and \(x\mapsto axb\) where \(a,b\) are \(C\)-algebraic elements in the symmetric Martindale ring of quotients of \(R\). As a corollary it is shown that the minimal polynomial of a \(C\)-algebraic derivation (resp. automorphism) takes the form \(\lambda^lh(\lambda)\) (resp. \((\lambda-1)^lh(\lambda)\)), where \(h(\lambda)\in C[\lambda]\) is such that \(h(0)\neq 0\) (resp. \(h(1)\neq 0\)) and \(l\) is odd if \(\text{char}(R)\neq 2\); if, however, \(\text{char}(R)=2\), then \(l\) is a \(2\)-power.
scientific article; zbMATH DE number 5151572