Show that $R$ is a Noetherian Ring if and only if $M_2(R)$ is a Noetherian ring.

Question

Let $R$ be a commutative ring with identity. Show that $R$ is a Noetherian Ring if and only if $M_2(R) = \{\text{2 x 2 matrices with coefficients in } R$} is a Noetherian ring.

I know that $R$ is Noetherian if any of the conditions hold:

• $R$ has the acending chain of ideals condition
• Every ideal in R is finitely generated
• Every nonempty set of partially ordered ideals has a maximal element.

I appreciate any help.