# Definition:Associative Algebra

## Definition

Let $R$ be a commutative ring.

Let $\left({A_R, *}\right)$ be an algebra over $R$.

Then $\left({A_R, *}\right)$ is an associative algebra if and only if $*$ is an associative operation.

That is:

$\forall a, b, c \in A_R: \left({a * b}\right) * c = a * \left({b * c}\right)$