Center (algebra)

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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. 
   ^ Kilp, 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. 
   


2. 
   ^ Ljapin, 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. 
   


3. 
   ^ Durbin, 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
   

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