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| In [[ring theory]], a branch of [[abstract algebra]], a '''quotient ring''', also known as '''factor ring''' or '''residue class ring''', is a construction quite similar to the [[factor group]]s of [[group theory]] and the [[quotient space (linear algebra)|quotient space]]s of [[linear algebra]].<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> One starts with a [[ring (mathematics)|ring]] ''R'' and a [[two-sided ideal]] ''I'' in ''R'', and constructs a new ring, the quotient ring ''R''/''I'', essentially by requiring that all elements of ''I'' be zero in ''R''. Intuitively, the quotient ring ''R''/''I'' is a "simplified version" of ''R'' where the elements of ''I'' are "ignored".
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| 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 of a ring|localization]].
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| ==Formal quotient ring construction==
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| Given a ring ''R'' and a two-sided ideal ''I'' in ''R'', we may define an [[equivalence relation]] ~ on ''R'' as follows:
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| :''a'' ~ ''b'' [[if and only if]] ''a'' − ''b'' is in ''I''.
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| Using the ideal properties, it is not difficult to check that ~ is a [[congruence relation]].
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| In case ''a'' ~ ''b'', we say that ''a'' and ''b'' are ''congruent [[Ideal (ring theory)|modulo]]'' ''I''.
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| The [[equivalence class]] of the element ''a'' in ''R'' is given by
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| : [''a''] = ''a'' + ''I'' := { ''a'' + ''r'' : ''r'' in ''I'' }.
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| This equivalence class is also sometimes written as ''a'' mod ''I'' and called the "residue class of ''a'' modulo ''I''".
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| 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
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| * (''a'' + ''I'') + (''b'' + ''I'') = (''a'' + ''b'') + ''I'';
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| * (''a'' + ''I'')(''b'' + ''I'') = (''a'' ''b'') + ''I''.
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| (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'').
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| 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'''''.
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| ==Examples==
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| *The quotient ''R''/{0} is [[naturally isomorphic]] to ''R'', and ''R''/''R'' is the [[zero ring]] {0}. This fits with the general rule of thumb that the larger the ideal ''I'', the smaller the quotient ring ''R''/''I''. If ''I'' is a proper ideal of ''R'', i.e., ''I'' ≠ ''R'', then ''R''/''I'' is not the zero ring.
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| *Consider the ring of [[integer]]s '''Z''' and the ideal of [[even number]]s, denoted by 2'''Z'''. Then the quotient ring '''Z'''/2'''Z''' has only two elements, zero for the even numbers and one for the odd numbers. It is naturally isomorphic to the [[finite field]] with two elements, '''F'''<sub>2</sub>. Intuitively: if you think of all the even numbers as 0, then every integer is either 0 (if it is even) or 1 (if it is odd and therefore differs from an even number by 1). [[Modular arithmetic]] is essentially arithmetic in the quotient ring '''Z'''/''n'''''Z''' (which has ''n'' elements).
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| *Now consider the ring '''R'''[''X''] of [[polynomial ring|polynomial]]s in the variable ''X'' with [[real number|real]] coefficients, and the ideal ''I'' = (''X''<sup>2</sup> + 1) consisting of all multiples of the polynomial ''X''<sup>2</sup> + 1. The quotient ring '''R'''[''X'']/(''X''<sup>2</sup> + 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''<sup>2</sup> + 1 = 0, i.e. ''X''<sup>2</sup> = −1, which is the defining property of ''i''.
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| *Generalizing the previous example, quotient rings are often used to construct [[field extension]]s. Suppose ''K'' is some [[field (mathematics)|field]] and ''f'' is an [[irreducible polynomial]] in ''K''[''X'']. Then ''L'' = ''K''[''X'']/(''f'') is a field whose [[minimal polynomial (field theory)|minimal polynomial]] over ''K'' is ''f'', which contains ''K'' as well as an element ''x'' = ''X'' + (''f'').
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| *One important instance of the previous example is the construction of the finite fields. Consider for instance the field '''F'''<sub>3</sub> = '''Z'''/3'''Z''' with three elements. The polynomial ''f''(''X'') = ''X''<sup>2</sup> + 1 is irreducible over '''F'''<sub>3</sub> (since it has no root), and we can construct the quotient ring '''F'''<sub>3</sub>[''X'']/(''f''). This is a field with 3<sup>2</sup>=9 elements, denoted by '''F'''<sub>9</sub>. The other finite fields can be constructed in a similar fashion.
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| *The [[coordinate ring]]s of [[algebraic variety|algebraic varieties]] are important examples of quotient rings in [[algebraic geometry]]. As a simple case, consider the real variety ''V'' = {(''x'',''y'') | ''x''<sup>2</sup> = ''y''<sup>3</sup> } as a subset of the real plane '''R'''<sup>2</sup>. The ring of real-valued polynomial functions defined on ''V'' can be identified with the quotient ring '''R'''[''X'',''Y'']/(''X''<sup>2</sup> − ''Y''<sup>3</sup>), and this is the coordinate ring of ''V''. The variety ''V'' is now investigated by studying its coordinate ring.
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| *Suppose ''M'' is a C<sup>∞</sup>-[[manifold]], and ''p'' is a point of ''M''. Consider the ring ''R'' = C<sup>∞</sup>(''M'') of all C<sup>∞</sup>-functions defined on ''M'' and let ''I'' be the ideal in ''R'' consisting of those functions ''f'' which are identically zero in some [[neighborhood (mathematics)|neighborhood]] ''U'' of ''p'' (where ''U'' may depend on ''f''). Then the quotient ring ''R''/''I'' is the ring of [[germ (mathematics)|germ]]s of C<sup>∞</sup>-functions on ''M'' at ''p''.
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| *Consider the ring ''F'' of finite elements of a [[hyperreal number|hyperreal field]] *'''R'''. It consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers ''x'' for which a standard integer ''n'' with −''n'' < ''x'' < ''n'' exists. The set ''I'' of all infinitesimal numbers in *'''R''', together with 0, is an ideal in ''F'', and the quotient ring ''F''/''I'' is isomorphic to the real numbers '''R'''. The isomorphism is induced by associating to every element ''x'' of ''F'' the [[standard part function|standard part]] of ''x'', i.e. the unique real number that differs from ''x'' by an infinitesimal. In fact, one obtains the same result, namely '''R''', if one starts with the ring ''F'' of finite hyperrationals (i.e. ratio of a pair of [[hyperinteger]]s), see [[construction of the real numbers]].
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| ===Alternative complex planes===
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| The quotients '''R'''[''X'']/(''X''), '''R'''[X]/(''X'' + 1), and '''R'''[''X'']/(''X'' − 1) are all isomorphic to '''R''' and gain little interest at first. But note that '''R'''[''X'']/(''X''<sup>2</sup>) is called the [[dual number]] plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of '''R'''[''X''] by ''X''<sup>2</sup>. This alternative complex plane arises as a [[subalgebra]] whenever the algebra contains a [[real line]] and a [[nilpotent]].
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| Furthermore, the ring quotient '''R'''[''X'']/(''X''<sup>2</sup> − 1) does split into '''R'''[''X'']/(''X'' + 1) and '''R'''[''X'']/(''X'' − 1), so this ring is often viewed as the [[Direct sum of algebras|direct sum]] '''R''' <math> \oplus</math> '''R'''.
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| Nevertheless, an alternative complex number ''z'' = ''x'' + ''y'' j is suggested by j as a root of X<sup>2</sup> − 1, compared to i as root of X<sup>2</sup> + 1 = 0. This plane of [[split-complex number]]s normalizes the direct sum <math>R \oplus R</math> by providing a basis {1, j } for 2-space where the identity of the algebra is at unit distance from the zero. With this basis a [[unit hyperbola]] may be compared to the [[unit circle]] of the [[complex plane|ordinary complex plane]].
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| ===Quaternions and alternatives===
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| Hamilton’s [[quaternion]]s of 1843 can be cast as '''R'''[''X'',''Y'']/(''X''<sup>2</sup> + 1, ''Y''<sup>2</sup> + 1, ''XY'' + ''YX''). If ''Y''<sup>2</sup> − 1 is substituted for ''Y''<sup>2</sup> + 1, then one obtains the ring of [[split-quaternion]]s. Substituting minus for plus in ''both'' the quadratic binomials also results in split-quaternions. The [[anti-commutative property]] YX = −XY implies that XY has for its square
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| : (''XY'')(''XY'') = ''X''(''YX'')''Y'' = −''X''(''XY'')''Y'' = − ''XXYY'' = −1.
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| The three types of [[biquaternion]]s can also be written as quotients by conscripting the three-indeterminate ring '''R'''[''X'',''Y'',''Z''] and constructing appropriate ideals.
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| ==Properties==
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| Clearly, if ''R'' is a [[commutative ring]], then so is ''R''/''I''; the converse however is not true in general.
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| The natural quotient map ''p'' has ''I'' as its [[kernel (algebra)|kernel]]; since the kernel of every ring homomorphism is a two-sided ideal, we can state that two-sided ideals are precisely the kernels of ring homomorphisms. | |
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| The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: ''the ring homomorphisms defined on R/I are essentially the same as the ring homomorphisms defined on R that vanish (i.e. are zero) on I''. More precisely: given a two-sided ideal ''I'' in ''R'' and a ring homomorphism ''f'' : ''R'' → ''S'' whose kernel contains ''I'', then there exists precisely one ring homomorphism ''g'' : ''R''/''I'' → ''S'' with ''gp'' = ''f'' (where ''p'' is the natural quotient map). The map ''g'' here is given by the well-defined rule ''g''([''a'']) = ''f''(''a'') for all ''a'' in ''R''. Indeed, this [[universal property]] can be used to ''define'' quotient rings and their natural quotient maps.
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| As a consequence of the above, one obtains the fundamental statement: every ring homomorphism ''f'' : ''R'' → ''S'' induces a [[ring isomorphism]] between the quotient ring ''R''/ker(''f'') and the image im(''f''). (See also: [[fundamental theorem on homomorphisms]].) | |
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| The ideals of ''R'' and ''R''/''I'' are closely related: the natural quotient map provides a [[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'' (i.e. ''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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| In [[commutative algebra]] and [[algebraic geometry]], the following statement is often used: If ''R'' ≠ {0} is a [[commutative]] ring and ''I'' is a [[maximal ideal]], then the quotient ring ''R''/''I'' is a [[field (mathematics)|field]]; if ''I'' is only a [[prime ideal]], then ''R''/''I'' is only an [[integral domain]]. A number of similar statements relate properties of the ideal ''I'' to properties of the quotient ring ''R''/''I''.
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| The [[Chinese remainder theorem]] states that, if the ideal ''I'' is the intersection (or equivalently, the product) of pairwise [[coprime#Generalizations|coprime]] ideals ''I<sub>1</sub>'',...,''I<sub>k</sub>'', then the quotient ring ''R''/''I'' is isomorphic to the [[product of rings|product]] of the quotient rings ''R''/''I<sub>p</sub>'', ''p''=1,...,''k''.
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| <!-- letter p (instead of, say, i) for readability -->
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| ==See also==
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| * [[Residue field]]
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| * [[Goldie's theorem]]
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| ==Notes==
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| {{reflist}}
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| ==Further references==
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| * F. Kasch (1978) ''Moduln und Ringe'', translated by DAR Wallace (1982) ''Modules and Rings'', [[Academic Press]], page 33.
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| * Neal H. McCoy (1948) ''Rings and Ideals'', §13 Residue class rings, page 61, Carus Mathematical Monographs #8, [[Mathematical Association of America]].
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| * {{cite book|author=Joseph Rotman | title =Galois Theory (2nd edition)| publisher=Springer|pages=21–3| year=1998 | isbn=0-387-98541-7}}
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| * [[B.L. van der Waerden]] (1970) ''Algebra'', translated by Fred Blum and John R Schulenberger, Frederick Ungar Publishing, New York. See Chapter 3.5, "Ideals. Residue Class Rings", pages 47 to 51.
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| ==External links==
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| * {{springer|title=Quotient ring|id=p/q076920}}
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| * [http://www.math.niu.edu/~beachy/aaol/rings.html#ideals Ideals and factor rings] from John Beachy's ''Abstract Algebra Online''
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| * {{planetmath reference|id=470|title=Quotient ring}}
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| [[Category:Ring theory]]
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