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.<ref name="KilpKilʹp2000">Template:Cite book</ref><ref name="Ljapin1968">Template:Cite book</ref>
• 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.<ref>Template:Cite book, Exercise 22.22</ref> 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)