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In Commutative algebra there is a notion of an integral extension: Let $P$ be a subring of $R$. Then $R$ is the integral extension of $P$ if each element of $R$ is a root of a monic polynomial with coefficients from $P$. Integral extensions have the following advantage: if $I$ is a maximal ideal of $R$, an integral extension of $P$, then $I\cap P$ is a maximal ideal of $P$.

My question:

Is there a similar notion for non-commutative (possibly unital) rings which gives a similar claim for maximal right/left ideals of the subring (say, intergrally embedded)?

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As far as I know these are normalizing extensions. –  user26857 Oct 31 '12 at 14:18
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