Maximal and prime ideals in matrix rings

Question

I know that if $I$ is an ideal in a ring $R$, then $M_n(I)$ is an ideal in $M_n(R)$. For example, the ideals in $\mathbb{Z}$ are $m\mathbb{Z}$ and so the ideals in $M_n(\mathbb{Z})$ are $M_n(m\mathbb{Z})$. The maximal (and prime) ideals in $\mathbb{Z}$ are $p\mathbb{Z}$ where $p$ is prime. Is this still true for $M_n(p\mathbb{Z})$? I mean, are $M_n(p\mathbb{Z})$ the maximal and prime ideals of $M_n(\mathbb{Z})$?

Answer 1

Yes, because this ideal correspondence between $R$ and $M_n(R)$ is containment preserving. That is, $M_n(I)\subseteq M_n(J)$ iff $I\subseteq J$. It also preserves ideal multiplication: $M_n(I)M_n(J)=M_n(IJ)$.

These allow you prove $M_n(I)$ is prime (resp. maximal) in $M_n(R)$ iff $I$ is prime (resp. maximal) in $R$.