Multiplicative semigroup automorphisms of upper triangular matrices over rings
Let \(R\) be a ring with \(1\) and \(C\) a central subring of \(R\). Let \(T_n(R)\) be the \(C\)-algebra of upper triangular \(n\times n\) matrices over \(R\). Several authors have characterized \(C\)-automorphisms of \(T_n(R)\). The authors generalize \textit{S.~Jøndrup}'s result [in Linear Algebra Appl. 221, 205-218 (1995; Zbl 0826.16034)]. The result obtained by Jøndrup says that if \(R\) is a semiprime ring or a ring in which all idempotents are central, then every \(C\)-automorphism of \(T_n(R)\) is the composite of an inner automorphism and an automorphism induced from a \(C\)-automorphism of \(R\). In this paper it is proved that if \(n\geqslant 2\) and \(R\) is a semiprime ring or a ring in which all idempotents are central, then \(f\) is a semigroup automorphism of \(T_n(R)\) if and only if there exist a nonsingular matrix \(P\) in \(T_n(R)\) and a ring automorphism \(\alpha\) of \(R\) such that \(f(A)=P^{-1}A^\alpha P\) for all \(A\) from \(T_n(R)\).
scientific article; zbMATH DE number 1354772