Let $ R $ to be a first Noetherian ring that is not necessarily commutative. Consider the two natural ways of "prolonging" $ R $: $ R[x]$ and $ M_n (R) $, polynomials on $ R $ and $ n $ by $ n $ dies on $ R $ respectively.
I'm interested in the behavior of the prime ideals as you extend $ R $. So my basic question is: if $ P $ is a prime localizable ideal of $ R $ (C & # 39; s, $ C (P) $ is (right and left) Ore), so is it true that $ P[x]$ is localizable in $ R[x]$ and $ M_n (P) $ is localizable in $ M_n (R) $?
This sounds both natural and perhaps obvious, but I can not find a way to approach it. Thank you for any help or references.