Center (algebra) explained

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)

Notes and References

1. Book: Kilp . Mati . Knauer . Ulrich . Mikhalev . Aleksandr V. . 2000 . Monoids, Acts and Categories . Walter de Gruyter . De Gruyter Expositions in Mathematics . 29 . 978-3-11-015248-7 . 25 .
2. Book: Ljapin, E. S. . A. A. Brown . J. M. Danskin . D. Foley . S. H. Gould . E. Hewitt . S. A. Walker . J. A. Zilber . 1968 . Semigroups . Translations of Mathematical Monographs . 3 . American Mathematical Soc. . Providence, Rhode Island . 978-0-8218-8641-0 . 96 .
3. Book: Durbin, John R. . Modern Algebra: An Introduction . 3rd . 1993 . John Wiley and Sons . 0-471-51001-7 . 118 . The center of a ring R is defined to be .., Exercise 22.22