# localization – Locating the prime ideals on the Noethean rings

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.