Associative algebra

Summary

An associative algebra over a commutative ring $K$ is a ring whose elements can also be multiplied by scalars from $K$. The scalar product is compatible with the ring product. Matrix algebras, polynomial algebras, group algebras and endomorphism algebras of vector spaces are the standard examples. The notion puts ring theory and linear algebra in one frame, so that dimension and bases are available when $K$ is a field. A commutative algebra over a ring is the special case where the ring product is commutative.