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# Center (algebra)

The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements.

• The center of a group G consists of all those elements x in G such that xg = gx for all g in G. This is a normal subgroup of G.

• The similarly named notion for a semigroup is defined likewise and it is a subsemigroup.[1][2]

• The center of a ring (or an associative algebra) R is the subset of R consisting of all those elements x of R such that xr = rx for all r in R.[3] The center is a commutative subring of R.

• The center of a Lie algebra L consists of all those elements x in L such that [x,a] = 0 for all a in L. This is an ideal of the Lie algebra L.

## See also

• Centralizer and normalizer

• Center (category theory)

## References

[1]
Citation Linkbooks.google.comKilp, Mati; Knauer, Ulrich; Mikhalev, Aleksandr V. (2000). Monoids, Acts and Categories. De Gruyter Expositions in Mathematics. 29. Walter de Gruyter. p. 25. ISBN 978-3-11-015248-7.
Sep 29, 2019, 7:36 AM
[2]
Citation Linkbooks.google.comLjapin, E. S. (1968). Semigroups. Translations of Mathematical Monographs. 3. Translated by A. A. Brown; J. M. Danskin; D. Foley; S. H. Gould; E. Hewitt; S. A. Walker; J. A. Zilber. Providence, Rhode Island: American Mathematical Soc. p. 96. ISBN 978-0-8218-8641-0.
Sep 29, 2019, 7:36 AM
[3]
Citation Linkopenlibrary.orgDurbin, John R. (1993). Modern Algebra: An Introduction (3rd ed.). John Wiley and Sons. p. 118. ISBN 0-471-51001-7. The center of a ring R is defined to be {cR: cr = rc for every rR}., Exercise 22.22
Sep 29, 2019, 7:36 AM
[4]
Citation Linkbooks.google.comMonoids, Acts and Categories
Sep 29, 2019, 7:36 AM
[5]
Citation Linkbooks.google.comSemigroups
Sep 29, 2019, 7:36 AM
[6]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 29, 2019, 7:36 AM