Definition:Algebra over Ring

Definition
An algebra over a ring $\left({G_R, \oplus}\right)$ is an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping $\oplus: G^2 \to G$.

It can be considered to be a generalization of an algebra over a field in which:
 * the vector space is replaced by a module
 * the field is replaced by a commutative ring.

Note that for this definition to be valid, it is important that $R$ be commutative, but it does not necessarily have to be a ring with unity.

Also see

 * Algebra over Field