Algebra over a field: Difference between revisions
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In [[abstract algebra]], an '''algebra over a field''' ''F'', or ''F''-'''algebra''' is a [[ring (mathematics)|ring]] ''A'' containing an [[field isomorphism|isomorphic]] copy of ''F'' in the [[centre of a ring|centre]]. Another way of expressing this is to say that ''A'' is a [[vector space]] over ''F'' equipped with a further [[algebraic structure]] of [[multiplication]] compatible with the vector space structure. | In [[abstract algebra]], an '''algebra over a field''' ''F'', or ''F''-'''algebra''' is a [[ring (mathematics)|ring]] ''A'' containing an [[field isomorphism|isomorphic]] copy of ''F'' in the [[centre of a ring|centre]]. Another way of expressing this is to say that ''A'' is a [[vector space]] over ''F'' equipped with a further [[algebraic structure]] of [[multiplication]] compatible with the vector space structure. | ||
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* Any [[extension field]] ''E''/''F'' can be regarded as an ''F''-algebra. | * Any [[extension field]] ''E''/''F'' can be regarded as an ''F''-algebra. | ||
* The [[matrix ring]] ''M''<sub>''n''</sub>(''F'') of ''n''×''n'' [[square matrix|square matrices]] with entries in ''F'' is an ''F''-algebra, with ''F'' embedded as the [[scalar matrix|scalar matrices]]. | * The [[matrix ring]] ''M''<sub>''n''</sub>(''F'') of ''n''×''n'' [[square matrix|square matrices]] with entries in ''F'' is an ''F''-algebra, with ''F'' embedded as the [[scalar matrix|scalar matrices]]. | ||
* The [[ | * The ring of [[quaternions]] '''H''' is a [[division ring]] with [[centre of a ring|centre]] the [[real number]]s '''R'''. It may thus be regarded as an '''R'''-algebra. | ||
==References== | ==References== | ||
*{{cite book | author=Serge Lang | authorlink=Serge Lang | title= | * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 }} | ||
Latest revision as of 15:55, 23 December 2008
In abstract algebra, an algebra over a field F, or F-algebra is a ring A containing an isomorphic copy of F in the centre. Another way of expressing this is to say that A is a vector space over F equipped with a further algebraic structure of multiplication compatible with the vector space structure.
Examples
- Any extension field E/F can be regarded as an F-algebra.
- The matrix ring Mn(F) of n×n square matrices with entries in F is an F-algebra, with F embedded as the scalar matrices.
- The ring of quaternions H is a division ring with centre the real numbers R. It may thus be regarded as an R-algebra.
References
- Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley. ISBN 0-201-55540-9.