Associative algebra: Difference between revisions

Revision as of 14:55, 8 June 2007

Definition

An associative algebra over a base ring $R$ is defined as a ring $A$ , along with the structure of a $R$ -module to $A$ .

In the particular case when $R$ and $A$ are both unital rings, this is equivalent to saying that we require an embedding of $R$ as a sub (unital ring) of $A$ .

We typically studiy algebras over a field, which are just vector spaces over the field equipped with a suitable compatible multiplication.

Sometimes, we also look at the non-associative notion of algebra, where we do not assume associativity of the multiplication for $A$ .

@@ Line 1: / Line 1: @@
 ==Definition==
-An algebra over a base ring <math>R</math> is defined as a ring <math>A</math>, along with the structure of a <math>R</math>-module to <math>A</math>.
+An '''associative algebra''' over a base ring <math>R</math> is defined as a ring <math>A</math>, along with the structure of a <math>R</math>-module to <math>A</math>.
 In the particular case when <math>R</math> and <math>A</math> are both unital rings, this is equivalent to saying that we require an embedding of <math>R</math> as a sub (unital ring) of <math>A</math>.
@@ Line 7: / Line 7: @@
 We typically studiy algebras over a field, which are just vector spaces over the field equipped with a suitable compatible multiplication.
-Sometimes, we also look at the ''non-associative'' notion of algebra, where we do not even assume associativity of the multiplication for <math>A</math>
+Sometimes, we also look at the ''non-associative'' notion of algebra, where we do not assume associativity of the multiplication for <math>A</math>.
 ==Related notions==