FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 3, PAGES 661-668

## On the general linear group over weak Noetherian associative algebras

I.Z.Golubchik

Let R be a weak Noetherian algebra with unity element over an infinite field, I an ideal in R, $n \geq 3$, En(R) the elementary subgroup in the general linear group GLn(R), En(R,I) the normal subgroup in En(R) generated by the elementary matrices $1 + \lambda e_{ij}$, $\lambda \in I$, $1 \leq i \neq j \leq n$, GLn(R,I) the kernel and Cn(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 GLn(R), then it is normalized by En(R) if and only if $E_n(R,F) \subseteq G \subseteq C_n(R,F)$ for some ideal F of R; [Cn(R,F),En(R)] = En(R,F) and in particular the groups En(R) and En(R,F) are normal in GLn(R) for all ideals F of R.

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