Let $R$ be a left Noetherian ($\mathbb N$-)graded ring and let $R_0$ be its $0$-th component. When $R$ is commutative it is well-known, and easy to prove, that the minimal prime ideals of $R$ are necessarily graded (see for example this question). What about the non-commutative case? Is it still true that minimal prime ideals are graded?
A similar question is the following. In the above situation, suppose that $R$ is flat as an $R_0$-module, are all the minimal prime ideals of $R$ of the form $R\otimes_{R_0}\frak p$ for some minimal prime ideal $\frak p$ of $R_0$.
If this is not true, is there any standard hypothesis to make it true?