# Classifying all commutative $\mathbb{R}$-algebras of matrices over $\mathbb{R}$?

I initially thought they were all isomorphic to some subring of the $$n \times n$$ diagonal matrices $$\mathcal{D} \cong \mathbb{R} \times \dots \times \mathbb{R}$$, but this was wrong: Every commutative ring of matrices over $\mathbb{R}$ is isomorphic to the diagonals?. One counterexample is matrices of the form (using block matrix notation) $$\begin{bmatrix} \alpha I_1 & A \\ 0 & \alpha I_{n-1} \\ \end{bmatrix}$$ for some $$1 \times (n-1)$$ real matrix block $$A$$ and some $$\alpha \in \mathbb{R}$$, which forms a commutative ring $$(\mathcal{U}, +, *)$$.

Are there other counterexamples? Can we classify all such rings up to isomorphism?

I use "ring" to mean "unital ring," but a similar classification for rngs would also be interesting.

From Unital rings within matrices, it seems that matrices in $$M_2(\mathbb{R})$$ of the form

$$\begin{bmatrix} a & b \\ -b & a-b \\ \end{bmatrix}$$

will be another example?

EDIT: If we require the commutative subring of $$M_n(\mathbb{R})$$ be closed under scalar multiplication, then it is a commutative $$\mathbb{R}$$-algebra.

• Do you require them to be closed under multiplication by scalars? If not then you can get many examples by restricting the coefficients e.g. to the rationals or integers. Commented Jun 20, 2020 at 6:44
• Yes, closed under scalar multiplication. Commented Jun 20, 2020 at 6:48
• @jskattt797 Then you should change the question to commutative algebras, not commutative rings. Commented Jun 20, 2020 at 8:14

This would be equivalent to classifying all commutative $$\mathbb R$$ algebras of dimension $$n$$. It’s a basic fact that every $$n$$ dimensional $$\mathbb R$$ algebra is isomorphic to a subring of $$M_n(\mathbb R)$$.
• Does this revelation give us any new information? If I understood correctly, the goal was already to classify all commutative matrix $\mathbb{R}$-algebras. Commented Jun 20, 2020 at 11:17
• We are trying to classify all commutative $n$-dimensional $\mathbb{R}$-algebras in $M_n(\mathbb{R})$ up to isomorphism. But all $n$-dimensional $\mathbb{R}$-algebras will be isomorphic to such an algebra, so there are at least as many commutative matrix $\mathbb{R}$-algebra isomorphism classes as there are commutative $\mathbb{R}$-algebra isomorphism classes in general (and at most as many). Is my understanding correct? Commented Jun 20, 2020 at 18:30