|
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra. One starts with a ring ''R'' and a two-sided ideal ''I'' in ''R'', and constructs a new ring, the quotient ring ''R''/''I'', whose elements are the cosets of ''I'' in ''R'' subject to special ''+'' and ''⋅'' operations. Quotient rings are distinct from the so-called 'quotient field', or field of fractions, of an integral domain as well as from the more general 'rings of quotients' obtained by localization. ==Formal quotient ring construction== Given a ring ''R'' and a two-sided ideal ''I'' in ''R'', we may define an equivalence relation ~ on ''R'' as follows: :''a'' ~ ''b'' if and only if ''a'' − ''b'' is in ''I''. Using the ideal properties, it is not difficult to check that ~ is a congruence relation. In case ''a'' ~ ''b'', we say that ''a'' and ''b'' are ''congruent modulo'' ''I''. The equivalence class of the element ''a'' in ''R'' is given by : () = ''a'' + ''I'' := . This equivalence class is also sometimes written as ''a'' mod ''I'' and called the "residue class of ''a'' modulo ''I''". The set of all such equivalence classes is denoted by ''R''/''I''; it becomes a ring, the factor ring or quotient ring of ''R'' modulo ''I'', if one defines * (''a'' + ''I'') + (''b'' + ''I'') = (''a'' + ''b'') + ''I''; * (''a'' + ''I'')(''b'' + ''I'') = (''a'' ''b'') + ''I''. (Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of ''R''/''I'' is (0 + ''I'') = ''I'', and the multiplicative identity is (1 + ''I''). The map ''p'' from ''R'' to ''R''/''I'' defined by ''p''(''a'') = ''a'' + ''I'' is a surjective ring homomorphism, sometimes called the ''natural quotient map'' or the ''canonical homomorphism''. 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「quotient ring」の詳細全文を読む スポンサード リンク
|