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{{short description|Reduction of a ring by one of its ideals}}
{{Ring theory sidebar|Basic}}

In [[ring theory]], a branch of [[abstract algebra]], a '''quotient ring''', also known as '''factor ring''', '''difference ring'''<ref>{{cite book | author-link=Nathan Jacobson | last1=Jacobson | first1=Nathan | title=Structure of Rings | publisher=American Mathematical Soc. | year=1984 | edition=revised | isbn=0-821-87470-5}}</ref> or '''residue class ring''', is a construction quite similar to the [[quotient group]] in [[group theory]] and to the [[quotient space (linear algebra)|quotient space]] in [[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 | author-link=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> It is a specific example of a [[quotient (universal algebra)|quotient]], as viewed from the general setting of [[universal algebra]]. Starting with a [[ring (mathematics)|ring]] <math>R</math> and a [[two-sided ideal]] <math>I</math> in {{tmath|1= R }}, a new ring, the quotient ring {{tmath|1= R\ /\ I }}, is constructed, whose elements are the [[cosets]] of <math>I</math> in <math>R</math> subject to special <math>+</math> and <math>\cdot</math> operations. (Quotient ring notation always uses a [[fraction slash]] "{{tmath|1= / }}".)

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]].

== Formal quotient ring construction ==
Given a ring <math>R</math> and a two-sided ideal <math>I</math> in {{tmath|1= R }}, we may define an [[equivalence relation]] <math>\sim</math> on <math>R</math> as follows:
: <math>a \sim b</math> [[if and only if]] <math>a - b</math> is in {{tmath|1= I }}.
Using the ideal properties, it is not difficult to check that <math>\sim</math> is a [[congruence relation]].
In case {{tmath|1= a \sim b }}, we say that <math>a</math> and <math>b</math> are ''congruent [[Ideal (ring theory)|modulo]]'' <math>I</math> (for example, <math>1</math> and <math>3</math> are congruent modulo <math>2</math> as their difference is an element of the ideal {{tmath|1= 2 \mathbb{Z} }}, the [[Parity (mathematics)|even integers]]). The [[equivalence class]] of the element <math>a</math> in <math>R</math> is given by:
<math display="block">\left[ a \right] = a + I := \left\lbrace a + r : r \in I \right\rbrace</math>

This equivalence class is also sometimes written as <math>a \bmod I</math> and called the "residue class of <math>a</math> modulo <math>I</math>".

The set of all such equivalence classes is denoted by {{tmath|1= R\ /\ I }}; it becomes a ring, the '''factor ring''' or '''quotient ring''' of <math>R</math> modulo {{tmath|1 = I }}, if one defines
* {{tmath|1= (a + I) + (b + I) = (a + b) + I }};
* {{tmath|1= (a + I)(b + I) = (ab) + I }}.
(Here one has to check that these definitions are [[well-defined]]. Compare [[coset]] and [[quotient group]].) The zero-element of <math>R\ /\ I</math> is {{tmath|1= \bar{0} = (0 + I) = I }}, and the multiplicative identity is {{tmath|1= \bar{1} = (1 + I) }}.

The map <math>p</math> from <math>R</math> to <math>R\ /\ I</math> defined by <math>p(a) = a + I</math> is a [[surjective]] [[ring homomorphism]], sometimes called the '''''natural quotient map''''' or the '''''[[canonical homomorphism]]'''''.

== Examples ==
* The quotient ring <math>R\ /\ \lbrace 0 \rbrace</math> is [[naturally isomorphic]] to {{tmath|1= R }}, and <math>R / R</math> is the [[zero ring]] {{tmath|1= \lbrace 0 \rbrace }}, since, by our definition, for any {{tmath|1= r \in R }}, we have that {{tmath|1= \left[ r \right] = r + R = \left\lbrace r + b : b \in R \right\rbrace }}, which equals <math>R</math> itself. This fits with the rule of thumb that the larger the ideal {{tmath|1= I }}, the smaller the quotient ring {{tmath|1= R\ /\ I }}. If <math>I</math> is a proper ideal of {{tmath|1= R }}, i.e., {{tmath|1= I \neq R }}, then <math>R / I</math> is not the zero ring.
* Consider the ring of [[integer]]s <math>\mathbb{Z}</math> and the ideal of [[even number]]s, denoted by {{tmath|1= 2 \mathbb{Z} }}. Then the quotient ring <math>\mathbb{Z} / 2 \mathbb{Z}</math> has only two elements, the coset <math>0 + 2 \mathbb{Z}</math> consisting of the even numbers and the coset <math>1 + 2 \mathbb{Z}</math> consisting of the odd numbers; applying the definition, {{tmath|1= \left[ z \right] = z + 2 \mathbb{Z} = \left\lbrace z + 2y : 2y \in 2\mathbb{Z} \right\rbrace }}, where <math>2 \mathbb{Z}</math> is the ideal of even numbers. It is naturally isomorphic to the [[finite field]] with two elements, {{tmath|F_{2} }}. Intuitively: if you think of all the even numbers as {{tmath|1= 0 }}, then every integer is either <math>0</math> (if it is even) or <math>1</math> (if it is odd and therefore differs from an even number by {{tmath|1= 1 }}). [[Modular arithmetic]] is essentially arithmetic in the quotient ring <math>\mathbb{Z} / n \mathbb{Z}</math> (which has <math>n</math> elements).
* Now consider the [[ring of polynomials]] in the variable <math>X</math> with [[real number|real]] [[coefficient]]s, {{tmath|1= \mathbb{R} [X] }}, and the ideal <math>I = \left( X^2 + 1 \right)</math> consisting of all multiples of the [[polynomial]] {{tmath|1= X^2 + 1 }}. The quotient ring <math>\mathbb{R} [X]\ /\ ( X^2 + 1 )</math> is naturally isomorphic to the field of [[complex number]]s {{tmath|1= \mathbb{C} }}, with the class <math>[X]</math> playing the role of the [[imaginary unit]] {{tmath|1= i }}. The reason is that we "forced" {{tmath|1= X^2 + 1 = 0 }}, i.e. {{tmath|1= X^2 = -1 }}, which is the defining property of {{tmath|1= i }}. Since any integer exponent of <math>i</math> must be either <math>\pm i</math> or {{tmath|1= \pm 1 }}, that means all possible polynomials essentially simplify to the form {{tmath|1= a + bi }}. (To clarify, the quotient ring {{tmath|1= \mathbb{R} [X]\ /\ ( X^2 + 1 ) }} is actually naturally isomorphic to the field of all linear polynomials {{tmath|1= aX + b; a,b \in \mathbb{R} }}, where the operations are performed modulo {{tmath|1= X^2 + 1 }}. In return, we have {{tmath|1= X^2 = -1 }}, and this is matching <math>X</math> to the imaginary unit in the isomorphic field of complex numbers.)
* Generalizing the previous example, quotient rings are often used to construct [[field extension]]s. Suppose <math>K</math> is some [[field (mathematics)|field]] and <math>f</math> is an [[irreducible polynomial]] in {{tmath|1= K[X] }}. Then <math>L = K[X]\ /\ (f)</math> is a field whose [[minimal polynomial (field theory)|minimal polynomial]] over <math>K</math> is {{tmath|1= f }}, which contains <math>K</math> as well as an element {{tmath|1= x = X + (f) }}.
* One important instance of the previous example is the construction of the finite fields. Consider for instance the field <math>F_3 = \mathbb{Z} / 3\mathbb{Z}</math> with three elements. The polynomial <math>f(X) = Xi^2 +1</math> is irreducible over <math>F_3</math> (since it has no root), and we can construct the quotient ring {{tmath|1= F_3 [X]\ /\ (f) }}. This is a field with <math>3^2 = 9</math> elements, denoted by {{tmath|1= F_9 }}. The other finite fields can be constructed in a similar fashion.
* 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 <math>V = \left\lbrace (x,y) | x^2 = y^3 \right\rbrace</math> as a subset of the real plane {{tmath|1= \mathbb{R}^2 }}. The ring of real-valued polynomial functions defined on <math>V</math> can be identified with the quotient ring {{tmath|1= \mathbb{R} [X,Y]\ /\ (X^2 - Y^3) }}, and this is the coordinate ring of {{tmath|1= V }}. The variety <math>V</math> is now investigated by studying its coordinate ring.
* Suppose <math>M</math> is a <math>\mathbb{C}^{\infty}</math>-[[manifold]], and <math>p</math> is a point of {{tmath|1= M }}. Consider the ring <math>R = \mathbb{C}^{\infty}(M)</math> of all <math>\mathbb{C}^{\infty}</math>-functions defined on <math>M</math> and let <math>I</math> be the ideal in <math>R</math> consisting of those functions <math>f</math> which are identically zero in some [[neighborhood (mathematics)|neighborhood]] <math>U</math> of <math>p</math> (where <math>U</math> may depend on {{tmath|1= f }}). Then the quotient ring <math>R\ /\ I</math> is the ring of [[germ (mathematics)|germs]] of <math>\mathbb{C}^{\infty}</math>-functions on <math>M</math> at {{tmath|1= p }}.
* Consider the ring <math>F</math> of finite elements of a [[hyperreal number|hyperreal field]] {{tmath|1= ^* \mathbb{R} }}. It consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers <math>x</math> for which a standard integer <math>n</math> with <math>-n < x < n</math> exists. The set <math>I</math> of all infinitesimal numbers in {{tmath|1= ^* \mathbb{R} }}, together with {{tmath|1= 0 }}, is an ideal in {{tmath|1= F }}, and the quotient ring <math>F\ /\ I</math> is isomorphic to the real numbers {{tmath|1= \mathbb{R} }}. The isomorphism is induced by associating to every element <math>x</math> of <math>F</math> the [[standard part function|standard part]] of {{tmath|1= x }}, i.e. the unique real number that differs from <math>x</math> by an infinitesimal.  In fact, one obtains the same result, namely {{tmath|1= \mathbb{R} }}, if one starts with the ring <math>F</math> of finite hyperrationals (i.e. ratio of a pair of [[hyperinteger]]s), see [[construction of the real numbers]].

=== Variations of complex planes ===
The quotients {{tmath|1= \mathbb{R} [X] / (X) }}, {{tmath|1= \mathbb{R} [X] / (X + 1) }}, and <math>\mathbb{R} [X] / (X - 1)</math> are all isomorphic to <math>\mathbb{R}</math> and gain little interest at first. But note that <math>\mathbb{R} [X] / (X^2)</math> is called the [[dual number]] plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of <math>\mathbb{R} [X]</math> by {{tmath|1= X^2 }}.  This variation of a complex plane arises as a [[subalgebra]] whenever the algebra contains a [[real line]] and a [[nilpotent]].

Furthermore, the ring quotient <math>\mathbb{R} [X] / (X^2 - 1)</math> does split into <math>\mathbb{R} [X] / (X + 1)</math> and {{tmath|1= \mathbb{R} [X] / (X - 1) }}, so this ring is often viewed as the [[Direct sum of algebras|direct sum]] {{tmath|1= \mathbb{R} \oplus \mathbb{R} }}.
Nevertheless, a variation on complex numbers <math>z = x + yj</math> is suggested by <math>j</math> as a root of {{tmath|1= X^2 - 1 = 0 }}, compared to <math>i</math> as root of {{tmath|1= X^2 + 1 = 0 }}. This plane of [[split-complex number]]s normalizes the direct sum <math>\mathbb{R} \oplus \mathbb{R}</math> by providing a basis <math>\left\lbrace 1, j \right\rbrace</math> 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]].

=== Quaternions and variations ===
Suppose <math>X</math> and <math>Y</math> are two non-commuting [[indeterminate (variable)|indeterminate]]s and form the [[free algebra]] {{tmath|1= \mathbb{R} \langle X, Y \rangle }}.  Then Hamilton's [[quaternion]]s of 1843 can be cast as:
<math display="block">\mathbb{R} \langle X, Y \rangle / ( X^2 + 1,\, Y^2 + 1,\, XY + YX )</math>

If <math>Y^2 - 1</math> is substituted for {{tmath|1= Y^2 + 1 }}, then one obtains the ring of [[split-quaternion]]s. The [[anti-commutative property]] <math>YX = -XY</math> implies that <math>XY</math> has as its square:
<math display="block">(XY) (XY) = X (YX) Y = -X (XY) Y = -(XX) (YY) = -(-1)(+1) = +1</math>

Substituting minus for plus in ''both'' the quadratic binomials also results in split-quaternions.

The three types of [[biquaternion]]s can also be written as quotients by use of the free algebra with three indeterminates <math>\mathbb{R} \langle X, Y, Z \rangle</math> and constructing appropriate ideals.

== Properties ==
Clearly, if <math>R</math> is a [[commutative ring]], then so is {{tmath|1= R\ /\ I }}; the converse, however, is not true in general.

The natural quotient map <math>p</math> has <math>I</math> 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.

The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: the ring homomorphisms defined on <math>R\ /\ I</math> are essentially the same as the ring homomorphisms defined on <math>R</math> that vanish (i.e. are zero) on {{tmath|1= I }}. More precisely, given a two-sided ideal <math>I</math> in <math>R</math> and a ring homomorphism <math>f : R \to S</math> whose kernel contains {{tmath|1= I }}, there exists precisely one ring homomorphism <math>g : R\ /\ I \to S</math> with <math>gp = f</math> (where <math>p</math> is the natural quotient map). The map <math>g</math> here is given by the well-defined rule <math>g([a]) = f(a)</math> for all <math>a</math> in {{tmath|1 R }}. Indeed, this [[universal property]] can be used to ''define'' quotient rings and their natural quotient maps.

As a consequence of the above, one obtains the fundamental statement: every ring homomorphism <math>f : R \to S</math> induces a [[ring isomorphism]] between the quotient ring <math>R\ /\ \ker (f)</math> and the image {{tmath|1= \mathrm{im} (f) }}. (See also: ''[[Fundamental theorem on homomorphisms]]''.)

The ideals of <math>R</math> and <math>R\ /\ I</math> are closely related: the natural quotient map provides a [[bijection]] between the two-sided ideals of <math>R</math> that contain <math>I</math> and the two-sided ideals of <math>R\ /\ I</math> (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 <math>M</math> is a two-sided ideal in <math>R</math> that contains {{tmath|1= I }}, and we write <math>M\ /\ I</math> for the corresponding ideal in <math>R\ /\ I</math> (i.e. {{tmath|1= M\ /\ I = p(M) }}), the quotient rings <math>R\ /\ M</math> and <math>(R / I)\ /\ (M / I)</math> are naturally isomorphic via the (well-defined) mapping {{tmath|1= a + M \mapsto (a + I) + M / I }}.

The following facts prove useful in [[commutative algebra]] and [[algebraic geometry]]: for <math>R \neq \lbrace 0 \rbrace</math> commutative, <math>R\ /\ I</math> is a [[Field (mathematics)|field]] if and only if <math>I</math> is a [[maximal ideal]], while <math>R / I</math> is an [[integral domain]] if and only if <math>I</math> is a [[prime ideal]]. A number of similar statements relate properties of the ideal <math>I</math> to properties of the quotient ring {{tmath|1= R\ /\ I }}.

The [[Chinese remainder theorem]] states that, if the ideal <math>I</math> is the intersection (or equivalently, the product) of pairwise [[coprime#Generalizations|coprime]] ideals {{tmath|1= I_1, \ldots, I_k }}, then the quotient ring <math>R\ /\ I</math> is isomorphic to the [[product of rings|product]] of the quotient rings {{tmath|1= R\ /\ I_n,\; n = 1, \ldots, k }}.

== For algebras over a ring ==
<!-- [[Quotient associative algebra]] redirects here -->
An [[associative algebra]] <math>A</math> over a [[commutative ring]] <math>R</math> is a ring itself. If <math>I</math> is an ideal in <math>A</math> (closed under <math>R</math>-multiplication), then <math>A / I</math> inherits the structure of an algebra over <math>R</math> and is the '''quotient algebra'''.

== See also ==
* [[Associated graded ring]]
* [[Residue field]]
* [[Goldie's theorem]]
* [[Quotient module]]

== Notes ==
{{reflist}}

== Further references ==
* F. Kasch (1978) ''Moduln und Ringe'', translated by DAR Wallace (1982) ''Modules and Rings'', [[Academic Press]], page 33.
* Neal H. McCoy (1948) ''Rings and Ideals'', §13 Residue class rings, page 61, Carus Mathematical Monographs #8, [[Mathematical Association of America]].
* {{cite book | author=Joseph Rotman | title=Galois Theory| publisher=Springer | pages=21–23 | year=1998 | isbn=0-387-98541-7 | edition=2nd }}
* [[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", pp. 47–51.

== External links ==
* {{springer|title=Quotient ring|id=p/q076920}}
* [http://www.math.niu.edu/~beachy/aaol/rings.html#ideals  Ideals and factor rings] from John Beachy's ''Abstract Algebra Online''

[[Category:Quotient objects|Ring]]
[[Category:Ring theory]]