Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ respectively.
I'm interested in the behavior of prime ideals as you extend $R$. So my basic question is this: if $P$ is a localizable prime ideal of $R$ (that is, $C(P)$ is (right and left) Ore), then is it true that $P[x]$ is localizable in $R[x]$ and $M_n(P)$ is localizable in $M_n(R)$?
This feels both natural and possibly obvious, but I can't work out how to approach it. Thanks for any help or references.
