Associative algebra: Difference between revisions
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==Definition== | ==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>. | 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>. | ||
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We typically studiy algebras over a field, which are just vector spaces over the field equipped with a suitable compatible multiplication. | 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 | 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== | ==Related notions== | ||
Latest revision as of 22:50, 7 May 2008
This article is about a standard (though not very rudimentary) definition in an area related to, but not strictly part of, group theory
Definition
An associative algebra over a base ring is defined as a ring , along with the structure of a -module to .
In the particular case when and are both unital rings, this is equivalent to saying that we require an embedding of as a sub (unital ring) of .
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 .