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!

  • 1
    $\begingroup$ No, and in general it is not. It is best to think of such a structure as a vecotorspace (and if your more inclined, a module) that just happens to have a notion of multiplication. Then the first axiom just expresses that the actions respect each other. An example you could look at are of-course function algebras, fields over their characteristic, or one of the classics, the quaternions (which are not commutative!). Algebras are actually very nice, for example, Commutative Rings are just $Z$-algebras (you can define algebras independent of rings, so this is actually meaning-ful). Hope this help $\endgroup$ – Pax Kivimae Jul 5 '14 at 13:29
  • $\begingroup$ Oh, also, lay off lang, get Artin, it's far better. $\endgroup$ – Pax Kivimae Jul 5 '14 at 13:30

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.


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