Let A and R be commutative rings with identity. Let \(E_ n(R)\) be the group generated by all elementary matrices \(e_{ij}(x)\), \(i\neq j\), \(x\in R\). Let \(GE_ n(R)\) denote the subgroup of \(GL_ n(R)\) generated by \(E_ n(R)\) and all diagonal matrices. The set of all maximal ideals of R is denoted by max(R). The triple (\(\phi\),\(\alpha\),\(\sigma)\) is called a system from A to R, where \(\phi\) is a bijection \(\phi\) : max(A)\(\to \max (R)\), \(J\mapsto M=\phi (J)\) with \(A/J=R/M\); \(\alpha\in R\) satisfies \(\alpha^ 2=1\) and \(2(\alpha -1)=0\); \(\sigma\) : \(A\to R\) is a bijection satisfying, for all \(x,y\in A\), \(x\in J\Leftrightarrow \sigma (x)\in M=\phi (J)\); \(\sigma (x)(\sigma (x)-1)(\alpha -1))=0\); \(\sigma (x+y)=\sigma (x)+\sigma (y)+\sigma (x)\sigma (y)(\alpha -1)\); \(\sigma (xy)=\sigma (x)\sigma (y).\)
The main result is that there is an exceptional isomorphism between \(E_ 3(A)\) and \(E_ 3(R)\Leftrightarrow there\) is a system (\(\phi\),\(\alpha\),\(\sigma)\) from A to R with \(\alpha\neq 1\). The authors also show that there is an isomorphism between \(GL_ 3({\mathbb{Z}}_ 4)\) and \(GL_ 3({\mathbb{F}}_ 2[x]/(x^ 2))\) even though \({\mathbb{Z}}_ 4\) and \({\mathbb{F}}_ 2[x]/(x^ 2)=\{0,1,x,1+x\}\) are not isomorphic. There are theorems exhibiting standard automorphisms of \(E_ 3(R)\) and \(GE_ 3(R)\) respectively.