For questions about algebras, their properties, and their structures. Use [tag:algebra-precalculus] or [tag:abstract-algebra] if your question is about algebra, not algebras.
An algebra over a field is a vector space equipped with a bilinear product. This product is not necessarily associative or unital, but if it is then the algebra is also a ring with unity. This can also be generalized by assuming that the scalars come from a commutative ring, rather than a field.
As with other algebraic objects, it is possible to define algebra homomorphisms, subalgebras, ideals, etc.
Examples
Group algebras, the algebra of polynomials $K[x]$ over a field $K$, and the quaternions are all associate algebras.
Every ring is an associative algebra over it's center.
The octonions are a non-associative algebra, and Lie algebras may not be associative.
Finite-dimensional algebras can be classified up to isomorphism by selecting a basis of $n$ and describing the multiplication of any two basis elements, which requires $n^3$ structure coefficients.
