Difference between revisions of "Central algebra"
From Encyclopedia of Mathematics
m (Corrected error: The tensor product of a CSA with a SA is _not_ central, except if the SA is actually a CSA too) |
(Category:Associative rings and algebras) |
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| − | An algebra with a unit element over a field, the centre of which (see [[Centre of a ring|Centre of a ring]]) coincides with the ground field. For example, the division ring of quaternions is a central algebra over the field of real numbers, but the field of complex numbers is not. | + | An algebra with a unit element over a field, the centre of which (see [[Centre of a ring|Centre of a ring]]) coincides with the ground field. For example, the division ring of quaternions is a central algebra over the field of real numbers, but the field of complex numbers is not. The full [[matrix algebra]] over a field is a central algebra. The tensor product of a simple algebra and a [[central simple algebra]] is a simple algebra, which is central if and only if the first one is. Every automorphism of a finite-dimensional central simple algebra is inner and its dimension is the square of an integer. |
====References==== | ====References==== | ||
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.S. Peirce, "Associative algebras" , Springer (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Deuring, "Algebren" , Springer (1935)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R.S. Peirce, "Associative algebras" , Springer (1980)</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939)</TD></TR> | ||
| + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Deuring, "Algebren" , Springer (1935)</TD></TR> | ||
| + | <TR><TD valign="top">[a4]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)</TD></TR> | ||
| + | <TR><TD valign="top">[a5]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
| + | |||
| + | [[Category:Associative rings and algebras]] | ||
Revision as of 19:49, 7 November 2014
An algebra with a unit element over a field, the centre of which (see Centre of a ring) coincides with the ground field. For example, the division ring of quaternions is a central algebra over the field of real numbers, but the field of complex numbers is not. The full matrix algebra over a field is a central algebra. The tensor product of a simple algebra and a central simple algebra is a simple algebra, which is central if and only if the first one is. Every automorphism of a finite-dimensional central simple algebra is inner and its dimension is the square of an integer.
References
| [1] | Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras" , Kiev (1980) (In Russian) |
| [2] | L.A. Skornyakov, "Elements of general algebra" , Moscow (1983) (In Russian) |
Comments
References
| [a1] | R.S. Peirce, "Associative algebras" , Springer (1980) |
| [a2] | A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) |
| [a3] | M. Deuring, "Algebren" , Springer (1935) |
| [a4] | I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) |
| [a5] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |
How to Cite This Entry:
Central algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_algebra&oldid=34318
Central algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_algebra&oldid=34318
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article