Associative algebra

Definition
An associative algebra over a base ring $$R$$ is defined as a ring $$A$$, along with the structure of a $$R$$-module to $$A$$.

In the particular case when $$R$$ and $$A$$ are both unital rings, this is equivalent to saying that we require an embedding of $$R$$ as a sub (unital ring) of $$A$$.

We typically studiy algebras over a field, which are just vector spaces over the field equipped with a suitable compatible multiplication.

Sometimes, we also look at the non-associative notion of algebra, where we do not assume associativity of the multiplication for $$A$$.

Related notions

 * Coalgebra
 * Bialgebra
 * Hopf algebra
 * Group algebra