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On normal subgroups of \(GL_ 2\) over rings with many units. (English) Zbl 0697.20040

The normal subgroup problem of linear groups over rings has received much attention and has been studied by many people. The present paper studies the case of two-dimensional linear groups, and proves the “sandwich theorem” holds under some hypotheses.
More precisely, assume A is an associative ring with 1 in which no proper one-sided ideal contains all u-1 where \(u\in GL_ 1A\), and every element of A is a sum of units. If furthermore one of the following three conditions is satisfied: (i) sr(A)\(\leq 1\), i.e., A satisfies the first Bass stable range condition; (ii) A/rad(A) is von Neumann regular where rad(A) is the Jacobson radical of A; (iii) for every element \(a\in A\) there is a finite sequence \(x_ 1,...,x_ N\) in A such that \(x_ 1+...+x_ N=1\) and \(1-ax_ i\in GL_ 1A\) for all i, then for every subgroup H of \(GL_ 2A\) which is normalized by \(GE_ 2A\) there is a unique ideal B of A such that \([E_ 2A,E_ 2B]\subset H\subset G_ 2(A,B)\) where \(G_ 2(A,B)\) is the inverse image of the center of \(GL_ 2(A/B)\) under the canonical homomorphism \(GL_ 2A\to GL_ 2(A/B)\).
Reviewer: Li Fuan

MSC:

20H25 Other matrix groups over rings
20E07 Subgroup theorems; subgroup growth
20G35 Linear algebraic groups over adèles and other rings and schemes
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