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. ↑ 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. https://books.google.com/books?id=4gPhmmW-EGcC&pg=PA25.
2. ↑ Ljapin, E. S. (1968). Semigroups. Translations of Mathematical Monographs. 3. Providence, Rhode Island: American Mathematical Soc.. p. 96. ISBN 978-0-8218-8641-0. https://books.google.com/books?id=G8pWKPp4tKwC&pg=PA96.
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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