I'm working on the following question.
Let $A$ be a finite dimensional semi-simple algebra over $\mathbb{C}$, and set $M_n(A)$ ring of $n$ by $n$ matrices over $A$.
(a) Show that $M_2(A)$ is semi-simple
(b) If $\dim_\mathbb{C}(A)$ is prime, show that $M_2(A)$ is not simple
(c) If $A$ is not commutative, there is a $t\in M_2(A)$ with $t^3\neq 0$ and $t^4 = 0$
For part a) I think we use Artin-Wedderburn to get $A=\oplus M_{n_i}(D_i)$. So $M_2(A) = \oplus M_{2n_i}(D_i)$. So $M_2(A)$ is semisimple
For part b) I don't see how prime is necessary. It feels like anything greater than 1 should give $M_2(A)$ is not simple because we can decompose $A$ into a direct sum
For part c) Not sure how to do this, I'm thinking maybe we can use the decomposition found in part a somehow?
a) Yes, if you have Artin-Wedderburn, then what you wrote is a way to see a matrix ring of matrix rings is just a bigger matrix ring, and the same holds for finite products of them.
b) Consider $A=M_2(\mathbb C)$, which has dimension $4$. $M_2(A)\cong M_4(\mathbb C)$ is simple. So... your feeling is not correct. In fact, $M_n(R)$ is simple iff $R$ is simple.
c) If $A$ is not commutative, then it contains somewhere a matrix ring which isn't trivial, and then $M_2(A)$ contains a matrix ring somewhere with side length at least $4$, when you expand the first matrix ring you found. So the task is to find a $4\times 4$ matrix with those properties, essentially.