Definition: A ring is Dedekind if every nonzero proper ideal factors into a product of prime ideals.
Let $R$ be a ring. $M_n(R)$ denotes the ring of matrices with elements from $R$. Is $M_n(R)$ Dedekind if $R$ is Dedekind?
I think the answer should be in some textbook, but I can't find it.
PS: I don't care the order of factors, so communitivity is not concerned.
If you are just referring to Dedekind domains, then of course $M_n(R)$ is never a domain for$n>1$. But the other important property (that $R$ is a hereditary ring) passes to $M_n(R)$.
Faith defined a Dedekind prime ring to be a hereditary, Noetherian, prime ring with no nontrivial idempotent ideals, and all those properties pass to the matrix ring, too. So the answer would be “yes” in that case.