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Recently, it has been shown that if $D$ is a finite dimensional
division ring then $GL_n(D)$ is not finitely generated $[2]$. Our
object here is to provide a general framework for the groups of
units of the left artinian rings. We prove that if $R$ is an
infinite $F$-algebra of finite dimension over $F$, then $U(R)$ is
not finitely generated. We show that none of infinite subnormal
subgroups of $GL_n(D)$ has finite maximal subgroup. Also in this
article, we prove that for any infinite left artinian ring $R$,
$U(R)$ has no finite maximal subgroup, a result is analogous to
one for rings $[6]$.
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