Isomorphisms of $\mathbb{R}[A]$ where $A$ is a $2\times2$ real matrix.

Question

I'm trying to answer a question in which I'm supposed to show that if $A$ is a $2\times2$ real matrix then $\mathbb{R}[A]$ (the polynomials in $A$ with real coefficients) is isomorphic to one of:

$$\mathbb{R},\space\space\space\mathbb{R}\times\mathbb{R}, \space\space\space\mathbb{R}[x]/\langle x^2 \rangle,\space\space\space\mathbb{C}$$

I've already shown, using the first isomorphism theorem, that $\mathbb{R}[A] \cong \frac{\mathbb{R}[x]}{\langle m(x) \rangle}$ where $m(x)$ is the minimal polynomial of $A$.

I can also see how, using this, I can show that if $m(x) = x -a$ for $a\in\mathbb{R}$ that we have $\mathbb{R}[A] \cong \mathbb{R}$ just by applying the first isomorphism theorem again and mapping $p(x) \in \mathbb{R}[A] $ to $p(a)$, but I'm at a loss with the other three cases.

I'd really appreciate whatever help you could give me.

Answer 1

You've already shown that $\Bbb{R}[A]\cong\Bbb{R}[x]/(m_A)$. So then the question becomes, what can $\Bbb{R}[x]/(m_A)$ be isomorphic to? Well, what do you know about $m_A$? It is the minimal polynomial of a real $2\times2$-matrix, so it is a monic polynomial of degree $1$ or $2$. As you say, you can show that if $\deg m_A=1$ then $\Bbb{R}[x]/(m_A)\cong\Bbb{R}$. Then it remains to see what the possible isomorphism types of $\Bbb{R}[x]/(m_A)$ are when $\deg m_A=2$. What can you say about the quotient ring $\Bbb{R}[x]/(x^2+ax+b)$? How does it depend on $a$ and $b$?

Answer 2

Since $A^2=tA-dI$, where $t$ and $d$ are the trace and determinant respectively, we know that $\mathbb{R}[A]$ is an $\mathbb{R}$-algebra with dimension at most $2$.

The case of dimension $1$ is when $A$ is scalar and this yields $\mathbb{R}[A]\cong\mathbb{R}$ (minimal polynomial of degree $1$).

The most interesting case is thus when it is $2$-dimensional. There are three subcases:

1. the minimal polynomial can be split as the product of two distinct degree one factors;
2. the minimal polynomial is of the form $(x-c)^2$;
3. the minimal polynomial is irreducible.

They correspond to the three algebras $\mathbb{R}\times\mathbb{R}$, $\mathbb{R}[x]/(x^2)$ and $\mathbb{C}$ respectively.