Derivations and right ideals of algebras.
Let \(R\) be a \(K\)-algebra acting densely on \(V_D\), where \(K\) is a commutative ring with unity and \(V\) is a right vector space over a division \(K\)-algebra \(D\). In this case, \(R\) is a prime ring, that is, \(aRb=0\), where \(a,b\in R\), implies that either \(a=0\) or \(b=0\). \(Q_{mr(R)}\) is the right maximal ring of quotients of \(R\) and let \(Q_s(R)\) denote the symmetric Martindale quotient of \(R\). Let \(\rho\) be a nonzero right ideal of \(R\) and let \(f(X_1,\dots,X_t)\) be a nonzero polynomial over \(K\) with constant term 0 such that \(\mu R\neq 0\) for some coefficient \(\mu\) of \(f(X_1,\dots,X_t)\). As an extended version of Theorem 1.1 of \textit{T.-K. Lee} and \textit{Y. Zhou} [Linear Algebra Appl. 431, No. 11, 2118-2126 (2009; Zbl 1183.15015)] in terms of derivations, the main theorem of the paper states that if \(d\colon R\to R\) is a nonzero derivation satisfying \(\text{rank\,}d(f(x_1,\dots,x_t))\leq m\) for all \(x_1,\dots,x_t\in\rho\) and for some positive integer \(m\), then either \(\rho\) is generated by an idempotent of finite rank or \(d=\text{ad}(b)\) for some \(b\in Q_s(R)\) of finite rank. In addition, if \(f(X_1,\dots,X_t)\) is multilinear, then \(b\) can be chosen such that \(\text{rank}(b)\leqslant 2(6t+13)m+2\). The key step is to prove the following result: if \(\text{rank}(d(R))=m\), where \(m\) is a positive integer, then either \(R\) is a simple Artinian ring or \(d=\text{ad}(b)\) for some \(b\in Q_s(R)\). The main theorem is proved using this result together with Theorem 1.1 of \textit{T.-K Lee, Y. Zhou} [loc. cit.].
right vector spaces over division algebras
scientific article; zbMATH DE number 5702820
scientific article