I am struggling to find a definition of an associative algebra.
Wikipedia (http://en.wikipedia.org/wiki/Associative_algebra) says
Let $R$ be a fixed commutative ring. An associative ''R''-algebra is an additive abelian group $A$ which has the structure of both a ring and an $R$-module in such a way that ring multiplication is $R$-bilinear:
$r\cdot(xy) = (r\cdot x)y = x(r\cdot y)$
for all $r ∈ R$ and $x, y ∈ A$. We say $A$ is unital if it contains an element 1 such that
$1 x = x = x 1$
for all $x ∈ A$. Note that such an element 1 must be unique if it exists at all.
If $A$ itself is commutative (as a ring) then it is called a commutative $R$-algebra.
I have also tried to find something in books as Serge Lang's Algebra and Undergraduate Algebra, but I could not find anything with respect to associative algebras.
Is multiplication of an associative algebra also commutative (because Wikipedia says "Let $R$ be a fixed commutative ring.")?
Thank you very much for your help!