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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?

jskattt797
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  • Let $k$ be a field, let $x_0, \ldots, x_n$ be indeterminates. The polynomial algebra $k[x_0, \ldots, x_n]$ over the variables $x_i$ is a commutative algebra. Also, if $I$ is an ideal in $k[x_0, \ldots, x_n]$, then also the quotient $k[x_0, \ldots, x_n]/I$ is a commutative algebra: in fact every finitely generated algebra is of this form (pretty much by definition). – Dunnò000 Jun 16 '20 at 16:20
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    Also note that your example, i.e. diagonal matrices, is better characterized as the algebra $\mathbb{R} \times \cdots \times \mathbb{R}$, with component wise operations. It is indeed a peculiar case of the above as it can be construed as the quotient $\mathbb{R}[x_1, \ldots, x_n]/I$ where $I$ is the ideal $I=(x_1+ \cdots +x_n-1) + (x_ix_j: 1 \le i,j \le n)$. – Dunnò000 Jun 16 '20 at 16:34
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    In general you may want to know that a quite standard definition of algebra over a field $k$ is just a ring $R$ together with a ring homomorphism $\varphi : k \rightarrow R$. – Dunnò000 Jun 16 '20 at 16:34

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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.

Matt
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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 $\mathbb C$.

Robert Israel
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