So an algebra $A$ over a field $K$ is a ring under the operations $+$ and $x$ and also a vector space under a scalar operation and the operation $+$.
What are some examples of algebras? When rings are introduced there’s plenty of trivial examples, but I can’t seem to find any simple examples on algebras?
Thank you.
Fields are algebras over themselves, and any ring containing a field in its center is an algebra over that field.
Then there's polynomial rings over fields, power series rings over fields, rings of functions on fields, matrix rings over fields, group rings over fields, direct products of any of the above (all being algebras over the common field.)
So even with just $\mathbb Q,\mathbb R,\mathbb C$, you should already be able to construct at least a dozen interestingly different algebras using it.
Actually, it is a thing that algebras are also defined over commutative rings. Doing it that way, every ring with characteristic $0$ and with identity is a $\mathbb Z$ algebra, and the ones with positive characteristic are algebras over $\mathbb Z_n$ for different $n$'s.