On the general linear group over weak Noetherian associative algebras

I. Z. Golubchik

Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 3, pp. 661-668

Résumé

Let $R$ be a weak Noetherian algebra with unity element over an infinite field, $I$ an ideal in $R$, $n\geq3$, $E_n(R)$ the elementary subgroup in the general linear group $GL_n(R)$, $E_n(R,I)$ the normal subgroup in $E_n(R)$ generated by the elementary matrices $1+\lambda e_{ij}$, $\lambda\in I$, $1\leq i\neq j\leq n$, $GL_n(R,I)$ the kernel and $C_n(R,I)$ the preimage of the center of the homomorphism $GL_n(R)\to GL_n(R/I)$ respectively. It is proved that if $G$ is a subgroup of $GL_n(R)$, then it is normalized by $E_n(R)$ if and only if $E_n(R,F)\subseteq G\subseteq C_n(R,F)$ for some ideal $F$ of $R$; $[C_n(R,F),E_n(R)]=E_n(R,F)$ and in particular the groups $E_n(R)$ and $E_n(R,F)$ are normal in $GL_n(R)$ for all ideals $F$ of $R$.