Are two matrices isomorphic? (as rings and as group)

Question

Assume that $M_2(R) , M_3(R)$ are matrices with real cells $2 \times 2$ , $3 \times 3$ respectively. 1)Are $M_2(R) , M_3(R)$ Isomorphic as rings under addition and multiplication ? why? 2) Are $M_2(R) , M_3(R)$ Isomorphic as additive groups? why? My thoughts: I think the part 1 can be done from some problems which already have answered in MSE. But for part 2 I don't know how to solve it I think the answer is not isomorphic but I don't know how to prove it. Could you please help me and answer that Please help me with your hints. Thanks

Answer 1

Hints:

1. There is an element $A \in M_3(\mathbb{R})$ such that $A^3 = 0$ and $A^2 \neq 0$, but in $M_2(\mathbb{R})$ there is no such element.

2. It's equivalent to the question whether $(\mathbb{R}^4, +)$ and $(\mathbb{R}^9, +)$ are isomorphic, because clearly

   $$\mathbb{R^4} \ni \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} \mapsto \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2(\mathbb{R})$$

   is an isomorphism and likewise $M_3(\mathbb{R}) \cong \mathbb{R}^9$.

   First try to solve a simpler problem: are $(\mathbb{R}, +)$ and $(\mathbb{R}^2, +)$ isomorphic?