Automorphisms and derivations of upper triangular matrix rings Original Research Article

Kezlan proved that for a commutative ring C, every C-automorphism of the ring of upper triangular matrices over C is inner. We generalize this result to rings in which all idempotents are central; moreover we show that for a semiprime ring A and central subring C, every C-automorphism of the ring of upper triangular matrices over C is the composite of an inner automorphism and an automorphism induced from a C-automorphism of A. By the method of proof we re-prove results of S. P. Coelho and C. P. Milies and of Mathis, stating that a derivation of a ring of upper triangular matrices of a C-algebra (n × n matrices over A) is a sum of an inner derivation and a derivation induced from a C-derivation of A. By an example we show that an extra assumption is needed for proving the above result of automorphisms of upper triangular matrices. Finally we consider automorphisms of subrings of n × n matrices over a commutative ring C, where entries over the diagonal are from C and below the diagonal are taken from a nil ideal. We prove that all such automorphisms are inner.