Extended Jacobson density theorem for graded rings with derivations and automorphisms. (Q543042)

The authors consider a group graded ring \(A\) with a graded simple left module \(M\). They define graded automorphisms of \(A\) and define derivations of \(A\) as certain mappings of \(A\) to endomorphisms of \(M\). Among a number of other results in the paper the authors extend density results for \(A\) acting on independent homogeneous elements of \(M\) to the case of a finite collection of nonisomorphic \(M_i\). The authors prove the equivalence of the existence of \(\{\alpha_1,\dots,\alpha_n\}\) of pairwise, independent, graded automorphisms of \(A\) (\(\alpha_i^{-1}\alpha_j\) is outer) to the following: for independent homogeneous elements \(\{x_1,\dots,x_k\}\) of \(M\) and any \(y_{ij}\in M\) there is \(a\in A\) so that \(a^{\alpha_i}x_j=y_{ij}\). A corresponding result holds for independent (modulo inner) derivations. The main result of the paper is similar to these, showing that a finite set \(\{d_i\}\) of independent derivations and a finite set \(\{\alpha_j\}\) of mutually outer automorphisms leads to a density result for \(\{a^{h_s}\}\) acting on independent homogeneous elements of \(M\), for some \(a\in A\), where \(h_s\) is a composition using some of the \(d_i\) followed by an \(\alpha_j\).