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.

References

↑ 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.
↑ 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.
↑ 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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[KilpKilʹp2000-1] 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.

[Ljapin1968-2] 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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Center (algebra)

See also

References