For questions about algebras, their properties, and their structures.

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.