Automorphisms of a certain subalgebra of the upper triangular matrix algebra

Abstract

For a commutative ring R with unity, the R-algebra of strictly upper triangular n x n matrices over R is denoted by N-n (R), where n is a positive integer greater than 1. For the identity matrix I, alpha epsilon R, A. N-n(R), the set of all elements alpha I+ A is defined as the scalar upper triangular matrix algebra STn (R) which is a subalgebra of the upper triangular matrices T-n(R). In this paper, we investigate the R-algebra automorphisms of STn (R). We extend the automorphisms of N-n (R) to STn (R) and classify all the automorphisms of STn (R). We generalize the results of Cao andWang and prove that not all automorphisms of STn (R) can be extended to the automorphisms of T-n(R).