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Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commutative operation, 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. * The center (ring theory), center of a ring (mathematics), 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''., Exercise 22.22 The center is a commutative ring, 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 (ring theory), ideal of the Lie algebra ''L''. See also *Centralizer and normalizer *Center (category theory) Refere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
