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Identities of a triangular extension. (Russian) Zbl 0536.16022

Let F be a field, let A and B be associative F-algebras with unit, let T(A) and T(B) be the T-ideals of identities of A and B in the free algebra \(F[x_ 1,...,x_ n]\) and let M be an (A,B)-bimodule. The algebra \(\begin{pmatrix} A & M \\ 0 & B \end{pmatrix}\) of triangular matrices \(\begin{pmatrix} \alpha & \mu \\ 0 & \beta \end{pmatrix}\), \(\alpha\in A\), \(\mu\in M\), \(\beta\in B\), is considered. Based on the methods of homological algebra and on the representation theory of algebras, the authors seek for conditions, under which the relation \(T\left(\begin{pmatrix} A & M \\ 0 & B \end{pmatrix}\right)=T(A)\cdot T(B)\) holds. In particular, they prove that this relation is true for all algebras A and B and for an (A,B)-bimodule M, which generates the category of (A,B)-bimodules.
Reviewer: U.Kaljulaid

MSC:

16Rxx Rings with polynomial identity
16Exx Homological methods in associative algebras
16S50 Endomorphism rings; matrix rings