Automorphism groups of generalized triangular matrix rings

作者：Anh P N; van Wyk L^*

来源：Linear Algebra and Its Applications, 2011, 434(4): 1018-1026.

DOI：10.1016/j.laa.2010.10.007

• Automorphism group
• Upper triangular matrix ring
• Strongly indecomposable ring
• Bimodule isomorphism

We call a ring strongly indecomposable if it cannot be represented as a non-trivial (i.e. M not equal 0) generalized triangular matrix ring ((R)(O) (M)(S)). for some rings R and S and some R-S-bimodule (R)M(S). Examples of such rings include rings with only the trivial idempotents 0 and 1, as well as endomorphism rings of vector spaces, or more generally, semiprime indecomposable rings. We show that if R and S are strongly indecomposable rings, then the triangulation of the non-trivial generalized triangular matrix ring ((R)(O) (M)(S)) is unique up to isomorphism; to be more precise, if phi : ((R)(O) (M)(S)) -> ((R')(O) (M')(S')) R -> R' and psi : S S' such that x := phi vertical bar(M) : M -> M' is an R-S-bimodule isomorphism relative to rho and psi. In particular, this result describes the automorphism groups of such upper triangular matrix ringsis an isomorphism, then there are isomorphisms rho : ((R)(O) (M)(S)).

• 出版日期2011-2-15