Show that the set of n dimensional matrices, where its components belong to a ring, is also a ring

Question

Where can I start? Can I prove for $2x2$ matrices and then prove the general statement using induction ?

Answer 1

I think it's a great idea to prove it for $2\times 2$ matrices first. However, there is no need for (explicit) induction. The general $n\times n$ proof is basically the same, only you use $\cdots$ a lot.