Prime ideals of the matrix ring

Question

I have to do the following question:

Given a unital ring $R$ and a natural number $n$ $\geq 1$, identify (with proof) the prime ideals of the matrix ring $\mathbb{M}_{n}(R)$.

I understand the theory, but putting it all together is proving difficult.

I understand that the prime ideals of $\mathbb{M}_{n}(R)$ should be $\mathbb{M}_{n}(\mathbb{P})$.

Can anyone suggest any steps as to how to prove this?

Answer 1

You just need to convince yourself that $M_n(A)M_n(B)=M_n(AB)$ for any two ideals $A,B$ of $R$.

After that it's just straightforward checking:

If $P$ is prime, then what does $M_n(A)M_n(B)\subseteq M_n(P)$ imply about $M_n(A)$ or $M_n(B)$?

If $M_n(P)$ is prime (and you don't know if $P$ is yet), then take two ideals $A,B$ and suppose $AB\subseteq P$ and ask what that means for $M_n(A)$ and $M_n(B)$. Let me know if you get stuck.