Representation of algebras by triangular matrices
Associative algebras are considered over a commutative ring \(\phi\) with identity. By a triangular system \(D\) of order \(n\) \((n\in N\cup \{\infty\})\) is understood a system of \(\phi\)-modules \(D_{ij}\) \((1\leq i\leq j\leq n+1)\) for which associative bilinear homomorphisms \(D_{ij}\times D_{jk}\to D_{ik}\) (i\(\leq j\leq k)\) are defined and \(D_{ii}\) are commutative algebras. The triangular system defines an algebra of triangular matrices \[ T_ m(D)=\begin{matrix} D_ 1 & D_{12} & D_{22} \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \end{matrix}. \] Some conditions for an embedding of such algebras in graded algebras of triangular matrices are found. An algebra \(R\) is called \(T_ n\)-algebra if it can be embedded in an algebra of triangular matrices of order \(n\) over a commutative algebra. The author proves, that if the commutant of \(R\), denoted by \((R,R)\) is a direct summand of R (as \(R\)-module), and \((R,R)^ 2=0\), then the algebra \(R\) is a \(T_ 2\)-algebra. For an algebra \(R\) to be a \(T_ 3\)-algebra the following condition is necessary: \[ (*)\quad \sum_{i}u_ iv_ i=0\quad \Rightarrow \quad \sum_{i}u_ ixv_ i=0\quad (u_ i,v_ i\in (R,R),\quad x\in R). \] If \((R,R)\) and \((R,R)^ 2\) are direct summands of \(R\), then the condition (*) is sufficient too.
scientific article; zbMATH DE number 3922895
scientific article