The group of automorphisms of certain subalgebras of matrix algebras
Let A be a prime ring which is also a K algebra, where K is a commutative ring, and denote by \(T_ n(A)\) the ring of upper triangular matrices over A. The author shows that every automorphism of \(T_ n(A)\) can be obtained by first applying an automorphism to A and then an inner automorphism. He goes on to study automorphisms of \(T_ n(a,R)\), the ring of \(n\times n\) matrices over a ring R whose entries below the main diagonal lie in the ideal a of R. If R is a (commutative) unique factorization domain and p is a prime element of R, then every automorphism of \(T_ n((p),R)\) is composed by an automorphism of R fixing (p), an inner automorphism and an element of the automorphism class group of \(T_ n((p),R)\) which is cyclic of order n. When (p) is replaced by a prime ideal of height \(>1\), every R automorphism is inner. With the help of the Jung-van der Kulk theorem [cf. \textit{P. M. Cohn}, Free rings and their relations (2nd Ed. Academic Press 1985; Zbl 0659.16001), p. 348], this gives a complete description of the automorphisms of \(T_ n((x,y),K[x,y])\), where K is a field and x, y indeterminates.
scientific article; zbMATH DE number 4204597