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词条 Center (algebra)
释义

  1. See also

  2. References

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. ^{{cite book |last1=Kilp |first1=Mati |last2=Knauer |first2=Ulrich |last3=Mikhalev |first3=Aleksandr V. |date=2000 |title=Monoids, Acts and Categories |publisher=Walter de Gruyter |series=De Gruyter Expositions in Mathematics |volume=29 |isbn=978-3-11-015248-7 |page=25 |url=https://books.google.com/books?id=4gPhmmW-EGcC&pg=PA25}}
2. ^{{cite book |last=Ljapin |first=E. S. |translator=A. A. Brown |translator2=J. M. Danskin |translator3=D. Foley |translator4=S. H. Gould |translator5=E. Hewitt |translator6=S. A. Walker |translator7=J. A. Zilber |date=1968 |title=Semigroups |series=Translations of Mathematical Monographs |volume=3 |publisher=American Mathematical Soc. |location=Providence, Rhode Island |isbn=978-0-8218-8641-0 |page=96 |url=https://books.google.com/books?id=G8pWKPp4tKwC&pg=PA96}}
3. ^{{cite book |last=Durbin |first=John R. |title=Modern Algebra: An Introduction |edition=3rd |date=1993 |publisher=John Wiley and Sons |isbn=0-471-51001-7 |page=118 |quote=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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1 : Abstract algebra
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