Automorphism groups of generalized triangular matrix rings.
A ring is called `strongly indecomposable' if it cannot be represented as a generalized triangular matrix ring \(\left(\begin{smallmatrix} R&M\\ 0&S\end{smallmatrix}\right)\) for some rings \(R\) and \(S\) and a nonzero \(R\)-\(S\)-bimodule \(_RM_S\). Typical examples are endomorphism rings of vector spaces and semiprime indecomposable rings. The main result of the paper states that if \(R\) and \(S\) are strongly indecomposable rings, then the triangulation of the nontrivial generalized matrix ring \(\left(\begin{smallmatrix} R&M\\ 0&S\end{smallmatrix}\right)\) is unique up to isomorphism. As a corollary, if \(A=\left(\begin{smallmatrix} R&M\\ 0&S\end{smallmatrix}\right)\), with \(M\neq 0\) and \(R\) and \(S\) strongly indecomposable, then \(\Aut(A)\) is a subgroup of \(\Aut(R)\times\Aut(S)\times M\times\Aut(M)\), consisting of all quadruples \((\rho,\psi,m,\chi)\), where \(\rho\in\Aut(R)\), \(\psi\in\Aut(S)\), \(m\in M\) and \(\chi\in\Aut(M)\) such that \(\chi\) is an \(R\)-\(S\)-bimodule automorphism of \(M\) relative to \(\rho\) and \(\psi\).
scientific article; zbMATH DE number 5848730