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 matrices over that share a common eigenbasis (in the case of all diagonal matrices, this eigenbasis forms ), though they need not have the same eigenvalues . Are there other examples?