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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク List of abstract algebra topics ： ウィキペディア英語版 List of abstract algebra topics Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called ''elementary algebra''. The distinction is rarely made in more recent writings. ==Basic language== Algebraic structures are defined primarily as sets with ''operations''. *Algebraic structure * *Subobjects: subgroup, subring, subalgebra, submodule etc. *Binary operation * *Closure of an operation * *Associative property * *Distributive property * *Commutative property *Unary operator * *Additive inverse, multiplicative inverse, inverse element * * *Identity element * * *Cancellation property *Finitary operation * *Arity Structure preserving maps called ''homomorphisms'' are vital in the study of algebraic objects. *Homomorphisms * *Kernels and cokernels * *Image and coimage * *Epimorphisms and monomorphisms * * *Isomorphisms * * * *Isomorphism theorems There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra. *Direct sum * *Direct limit *Direct product * *Inverse limit *Quotient objects: quotient group, quotient ring, quotient module etc. *Tensor product Advanced concepts: *Category theory * *Category of groups * * *Category of abelian groups * *Category of rings * *Category of modules * * *Morita equivalence, Morita duality * *Category of vector spaces *Homological algebra * *Filtration (algebra) * *Exact sequence * *Functor *Zorn's lemma 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of abstract algebra topics」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.