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Center (algebra)

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 {c ∈ R: cr = rc for every r ∈ R}., 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