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EspañolCENTER (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. 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 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''.
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