algebra (module) (original) (raw)

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

which is R-bilinearPlanetmathPlanetmath.

1 Unital associative algebras

In these cases, the “product” (as it is called) of two elementsv and w of the module, is denoted simply by v⁢w or v∙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:

2 Lie algebras

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

[v,[w,x]]+[w,[x,v]]+[x,[v,w]]=0⁢ for all ⁢v,w,x∈M

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

[v,w]+[w,v]=0⁢ for all ⁢v,w∈M

for any Lie algebra M.

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