Axioms of associative algebra?

Question

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!

Answer 1

We can reexpress the axioms you gave this way:

1. A is an associative ring
2. A is an R module for a commutative ring R
3. The ring product is compatible with the module product

These three things describe an R-algebra A.

R being commutative does not imply A is also commutative. A good first example is a matrix ring over a field F. For n by n matrices, n>2, the matrix ring is noncommutative even though it is an F algebra.