Associative algebra: Difference between revisions

Revision as of 14:51, 8 June 2007

Definition

An 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 even assume associativity of the multiplication for $A$

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 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>.
+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>
 ==Related notions==