Are the rings $\mathbb{R}^2$ and $\mathbb{R}^3$ isomorphic?

Question

Are the rings $\mathbb{R}^2$ and $\mathbb{R}^3$ isomorphic, where $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$ is the set of all pairs $(a,b)$ with $a,b \in \mathbb{R}$, and $\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ is the set of all triples $(a,b,c)$ with $a,b,c \in \mathbb{R}$, using component wise addition and multiplication?

Answer 1

HINT. If they are going to be isomorphic, it should work for any ring. Think of the special case where the ring is actually a field, i.e. $R=\mathbb{k}$. Can you think of some theorems from basic Linear Algebra to say whether or not the rings $R^2=\mathbb{k}^2$ and $R^3=\mathbb{k}^3$ are isomorphic? Maybe focus on the only difference between them...

Answer 2

Consider idempotents in each ring, that is solutions of $e^2=e$. In your first ring you are solving $(a^2,b^2)=(a,b)$, and in the second, $(a^2,b^2,c^2)=(a,b,c)$. How may solutions have you in each case?