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. More specifically:

• 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 center of a ring R is the subset of R consisting of all those elements x of R such that xr = rx for all r in R. The center is a commutative subring of R, so R is an algebra over its center.
• The center of an algebra A consists of all those elements x of A such that xa = ax for all a in A. See also: central simple algebra.
• 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.
• The center of a monoidal category C consists of pairs (A,u) where A is an object of C, and a natural isomorphism satisfying certain axioms.