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Extended Jacobson density theorem for graded rings with derivations and automorphisms. (English) Zbl 1232.16034

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 \(\alpha_j\).

MSC:

16W50 Graded rings and modules (associative rings and algebras)
16W25 Derivations, actions of Lie algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16N20 Jacobson radical, quasimultiplication
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