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On normal subgroups of $GL\sb 2$ 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 $u\in GL\sb 1A$, and every element of A is a sum of units. If furthermore one of the following three conditions is satisfied: (i) sr(A)$\le 1$, 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 $x\sb 1,...,x\sb N$ in A such that $x\sb 1+...+x\sb N=1$ and $1-ax\sb i\in GL\sb 1A$ for all i, then for every subgroup H of $GL\sb 2A$ which is normalized by $GE\sb 2A$ there is a unique ideal B of A such that $[E\sb 2A,E\sb 2B]\subset H\subset G\sb 2(A,B)$ where $G\sb 2(A,B)$ is the inverse image of the center of $GL\sb 2(A/B)$ under the canonical homomorphism $GL\sb 2A\to GL\sb 2(A/B)$.
Reviewer: Li Fuan

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