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Normal subgroups of the general linear groups over von Neumann regular rings. (English) Zbl 0594.16007
For a ring A which is von Neumann regular modulo its Jacobson radical, subgroups of $GL\sb n(A)$ normalized by the subgroup $E\sb n(A)$ of elementary matrices are characterized. Namely, assuming $n\ge 3$, a subgroup H of $GL\sb n(A)$ is normalized by $E\sb n(A)$ if and only if there is an ideal B of A (necessarily unique) such that $E\sb n(A,B)\subseteq H\subseteq G\sb n(A,B)$, where $G\sb n(A,B)$ is the inverse image of the center of $GL\sb n(A/B)$ and $E\sb n(A,B)$ is the normal subgroup of $E\sb n(A)$ generated by all elementary matrices in $G\sb n(A,B)$. This generalizes the classical characterization of normal subgroups of $GL\sb n$ of a field obtained by {\it L. E. Dickson} [Trans. Am. Math. Soc. 2, 363-394 (1901)] and extended to $GL\sb n$ of certain other rings by later authors.
Reviewer: K.R.Goodearl

16E50Von Neumann regular rings and generalizations
20H25Other matrix groups over rings
20E07Subgroup theorems; subgroup growth
15A30Algebraic systems of matrices
16S50Endomorphism rings: matrix rings
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