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Quotient rings are unrelated to the quotient field, or Field Of Fractions , of an Integral Domain , and also unrelated to the rings of quotients resulting from Localization Of Rings . 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 ''b'' − ''a'' 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 : {Link without Title} = ''a'' + ''I'' := { ''a'' + ''r'' : ''r'' in ''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
(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'''''. EXAMPLES The most extreme examples of quotient rings are provided by Modding Out the most extreme ideals, {0} and R itself. R/{0} is Naturally Isomorphic to R, and R/R is the Trivial Ring {0}. This fits with the general rule of thumb that ''the smaller the ideal I, the larger the quotient ring R/I''. If ''I'' is a proper ideal of ''R'', i.e. ''I'' ≠ ''R'', then ''R''/''I'' won't be the trivial ring. Consider the ring of is essentially arithmetic in the quotient ring Z/''n''Z (which has ''n'' elements). Now consider the ring R of Polynomial s in the variable ''X'' with Real coefficients, and the ideal ''I'' = (''X''2 + 1) consisting of all multiples of the polynomial ''X''2 + 1. The quotient ring R[''X'' /(''X''2 + 1) is naturally isomorphic to the field of Complex Number s '''C''', with the class [''X''] playing the role of the Imaginary Unit ''i''. The reason: we "forced" ''X''2 + 1 = 0, i.e. ''X''2 = -1, which is the defining property of ''i''. Generalizing the previous example, quotient rings are often used to construct Field Extension s. Suppose ''K'' is some Field and ''f'' is an Irreducible Polynomial in ''K'' Then ''L'' = ''K''[''X'' /(''f'') is a field which contains ''K'' as well as an element ''x'' = ''X'' + (''f'') whose Minimal Polynomial over ''K'' is ''f''. One important instance of the previous example is the construction of the Finite Field s. Consider for instance the field F3 = '''Z'''/3'''Z''' with three elements. The polynomial ''f''(''X'') = ''X''2 + 1 is irreducible over F3 (since it has no root), and we can construct the quotient ring F3 {Link without Title} /(''f''). This is a field with 32=9 elements, denoted by F9. The other finite fields can be constructed in a similar fashion. | ||
| The Ideals Of ''R'' And ''R''/''I'' Are Closely Related: The Natural Quotient Map Provides A | "http://wwwinformationdelightinfo/encyclopedia/entry/bijection" class="copylinks">Bijection between the two-sided ideals of ''R'' that contain ''I'' and the two-sided ideals of ''R''/''I'' (the same is true for left and for right ideals) This relationship between two-sided ideal extends to a relationship between the corresponding quotient rings: if ''M'' is a two-sided ideal in ''R'' that contains ''I'', and we write ''M''/''I'' for the corresponding ideal in ''R''/''I'' (ie ''M''/''I'' = ''p''(''M'')), the quotient rings ''R''/''M'' and (''R''/''I'')/(''M''/''I'') are naturally isomorphic via the (well-defined!) mapping ''a'' + ''M'' -> (''a''+''I'') + ''M''/''I'' |
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