Associative algebras with commutative multiplication?

Question

I.e. the bilinear map/product is not only associative, but also commutative. I am looking for examples of unital associative algebras, so they should be a vector space and a ring, not a vector space and a rng.

One example, inspired by When is matrix multiplication commutative?, is the set of all diagonal matrices. Generalizing this, we could look at all $n \times n$ matrices over $\mathbb{R}$ that share a common eigenbasis (in the case of all diagonal matrices, this eigenbasis forms $I_n$), though they need not have the same eigenvalues $\in \mathbb{R}$. Are there other examples?

Answer 1

Another class of examples: continuous $\mathbb C$-valued functions on some topological space. And various subalgebras of that where some restrictions are placed on the functions, e.g. analytic functions on some domain in $\mathbb C$.

Answer 2

Yes, lots.

Take for example the polynomial algebra $\mathbb{k}[x_1,...,x_n]$ for a field $\mathbb{k}$.

The complex numbers in particular are a commutative, associative algebra over $\mathbb{R}$. The quaternions aren't though, because they're not commutative.

Subalgebras of both your examples and mine are also examples.