# algebra (module)

Given a commutative ring $R$, an algebra over $R$ is a module $M$ over $R$, endowed with a law of composition

 $f:M\times M\to M$

which is $R$-bilinear.

Most of the important algebras in mathematics belong to one or the other of two classes: the unital associative algebras, and the Lie algebras.

## 1 Unital associative algebras

In these cases, the “product” (as it is called) of two elements $v$ and $w$ of the module, is denoted simply by $vw$ or $v\centerdot w$ or the like.

Any unital associative algebra is an algebra in the sense of djao (a sense which is also used by Lang in his book Algebra (Springer-Verlag)).

Examples of unital associative algebras:

tensor algebras and quotients of them

Cayley algebras, such as the ring of quaternions

– the ring of endomorphisms of a vector space, in which the bilinear product of two mappings is simply the composite mapping.

## 2 Lie algebras

In these cases the bilinear product is denoted by $[v,w]$, and satisfies

 $[v,v]=0\textrm{ for all }v\in M$
 $[v,[w,x]]+[w,[x,v]]+[x,[v,w]]=0\textrm{ for all }v,w,x\in M$

The second of these formulas is called the Jacobi identity. One proves easily

 $[v,w]+[w,v]=0\textrm{ for all }v,w\in M$

for any Lie algebra M.

Lie algebras arise naturally from Lie groups, q.v.

Title algebra (module) Algebramodule 2013-03-22 13:20:50 2013-03-22 13:20:50 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 13B99 msc 20C99 msc 16S99 Jacobi identity