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Suppose that $D$ is a division ring with center $F$ and $N$ is a
non-central normal subgroup of $GL_n(D)$. In this paper we
generalize some known results about maximal subgroups of $GL_n(D)$
to maximal subgroups of $N$. More precisely we prove that if $M$
is a maximal subgroup of $N$ such that $F[M]$ satisfies a
polynomial identity and $CM_n(D)(M)\backslash F$ contains an
algebraic element over $F$ or $CM_n(D)(M) = F$ and either $n \geq
2$ or $n = 1$ and $M$ is not abelian, then $[D: F] < \infty$.
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