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