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algebra (module)
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(Definition)
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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.
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
- polynomial rings
- the ring of endomorphisms of a vector space, in which the bilinear product of two mappings is simply the composite mapping.
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.
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Jacobi identity |
This object's parent.
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Cross-references: Lie groups, formulas, composite, mappings, product, bilinear, vector space, ring of endomorphisms, polynomial rings, quaternions, ring, Cayley algebras, quotients, tensor algebras, Lie algebras, associative, unital, classes, algebras, composition, module, algebra, commutative ring
There are 8 references to this entry.
This is version 2 of algebra (module), born on 2002-12-31, modified 2005-04-14.
Object id is 3865, canonical name is AlgebraModule.
Accessed 6883 times total.
Classification:
| AMS MSC: | 16S99 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Miscellaneous) | | | 20C99 (Group theory and generalizations :: Representation theory of groups :: Miscellaneous) | | | 13B99 (Commutative rings and algebras :: Ring extensions and related topics :: Miscellaneous) |
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Pending Errata and Addenda
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