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On the structure of $GL\sb 2$ over stable range one rings. (English) Zbl 0702.20037
For a large class of rings R (including all commutative rings) Vaserstein has proved that, when $n\ge 3$, the subgroups of $GL\sb n(R)$ normalized by $E\sb n(R)$, the subgroup generated by the elementary matrices, are completely determined by the R-ideals. From the extremely complicated normal subgroup structure of $SL\sb 2({\bbfZ})$, where ${\bbfZ}$ is the ring of rational integers, it is clear that for a general commutative ring R the subgroups of $GL\sb 2(R)$, normalized by $E\sb 2(R)$, cannot be completely determined in this way by special subsets of R like, for example, R-ideals. For anything like Vaserstein’s result to extend to the case $n=2$ it would appear that the ring has to contain “sufficiently many units”. Costa and Keller have proved that, if A is a commutative $SR\sb 2$-ring containing 1/2, then the normal subgroups of $SL\sb 2(A)$ are completely determined by the A-ideals. (Fields and semi-local rings, for example, are $SR\sb 2$-rings.) In the present paper the authors have extended this result to all $SR\sb 2$-rings B containing 1/2. They prove that the subgroups of $GL\sb 2(B)$, normalized by $E\sb 2(B)$, are completely determined by special subsets of B called quasi-ideals. (In any commutative ring containing 1/2 any quasi-ideal is an ideal.)
Reviewer: A.W.Mason

20H25Other matrix groups over rings
20E07Subgroup theorems; subgroup growth
20G35Linear algebraic groups over adèles and other rings and schemes
Full Text: DOI
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