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Quotient ring

## Summary

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring[1] or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra.[2][3] It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring ${\displaystyle R}$ and a two-sided ideal ${\displaystyle I}$ in ${\displaystyle R}$, a new ring, the quotient ring ${\displaystyle R/I}$, is constructed, whose elements are the cosets of ${\displaystyle I}$ in ${\displaystyle R}$ subject to special ${\displaystyle +}$ and ${\displaystyle \cdot }$ operations. (Quotient ring notation always uses a fraction slash "/".)

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 ${\displaystyle R}$  and a two-sided ideal ${\displaystyle I}$  in ${\displaystyle R}$ , we may define an equivalence relation ${\displaystyle \sim }$  on ${\displaystyle R}$  as follows:

${\displaystyle a\sim b}$  if and only if ${\displaystyle a-b}$  is in ${\displaystyle I}$ .

Using the ideal properties, it is not difficult to check that ${\displaystyle \sim }$  is a congruence relation. In case ${\displaystyle a\sim b}$ , we say that ${\displaystyle a}$  and ${\displaystyle b}$  are congruent modulo ${\displaystyle I}$  (for example, ${\displaystyle 1}$  and ${\displaystyle 3}$  are congruent modulo ${\displaystyle 2}$  as their difference is an element of the ideal ${\displaystyle 2\mathbb {Z} }$ , the even integers). The equivalence class of the element ${\displaystyle a}$  in ${\displaystyle R}$  is given by: ${\displaystyle \left[a\right]=a+I:=\left\lbrace a+r:r\in I\right\rbrace }$

This equivalence class is also sometimes written as ${\displaystyle a{\bmod {I}}}$  and called the "residue class of ${\displaystyle a}$  modulo ${\displaystyle I}$ ".

The set of all such equivalence classes is denoted by ${\displaystyle R/I}$ ; it becomes a ring, the factor ring or quotient ring of ${\displaystyle R}$  modulo ${\displaystyle I}$ , if one defines

• ${\displaystyle (a+I)+(b+I)=(a+b)+I}$ ;
• ${\displaystyle (a+I)(b+I)=(ab)+I}$ .

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of ${\displaystyle R/I}$  is ${\displaystyle {\bar {0}}=(0+I)=I}$ , and the multiplicative identity is ${\displaystyle {\bar {1}}=(1+I)}$ .

The map ${\displaystyle p}$  from ${\displaystyle R}$  to ${\displaystyle R/I}$  defined by ${\displaystyle p(a)=a+I}$  is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism.

## Examples

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

### Variations of complex planes

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

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

### Quaternions and variations

Suppose ${\displaystyle X}$  and ${\displaystyle Y}$  are two non-commuting indeterminates and form the free algebra ${\displaystyle \mathbb {R} \langle X,Y\rangle }$ . Then Hamilton's quaternions of 1843 can be cast as: ${\displaystyle \mathbb {R} \langle X,Y\rangle /(X^{2}+1,\,Y^{2}+1,\,XY+YX)}$

If ${\displaystyle Y^{2}-1}$  is substituted for ${\displaystyle Y^{2}+1}$ , then one obtains the ring of split-quaternions. The anti-commutative property ${\displaystyle YX=-XY}$  implies that ${\displaystyle XY}$  has as its square: ${\displaystyle (XY)(XY)=X(YX)Y=-X(XY)Y=-(XX)(YY)=-(-1)(+1)=+1}$

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

The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminates ${\displaystyle \mathbb {R} \langle X,Y,Z\rangle }$  and constructing appropriate ideals.

## Properties

Clearly, if ${\displaystyle R}$  is a commutative ring, then so is ${\displaystyle R/I}$ ; the converse, however, is not true in general.

The natural quotient map ${\displaystyle p}$  has ${\displaystyle I}$  as its 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 ${\displaystyle R/I}$  are essentially the same as the ring homomorphisms defined on ${\displaystyle R}$  that vanish (i.e. are zero) on ${\displaystyle I}$ . More precisely, given a two-sided ideal ${\displaystyle I}$  in ${\displaystyle R}$  and a ring homomorphism ${\displaystyle f:R\to S}$  whose kernel contains ${\displaystyle I}$ , there exists precisely one ring homomorphism ${\displaystyle g:R/I\to S}$  with ${\displaystyle gp=f}$  (where ${\displaystyle p}$  is the natural quotient map). The map ${\displaystyle g}$  here is given by the well-defined rule ${\displaystyle g([a])=f(a)}$  for all ${\displaystyle a}$  in ${\displaystyle R}$ . 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 ${\displaystyle f:R\to S}$  induces a ring isomorphism between the quotient ring ${\displaystyle R/\ker(f)}$  and the image ${\displaystyle \mathrm {im} (f)}$ . (See also: Fundamental theorem on homomorphisms.)

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

The following facts prove useful in commutative algebra and algebraic geometry: for ${\displaystyle R\neq \lbrace 0\rbrace }$  commutative, ${\displaystyle R/I}$  is a field if and only if ${\displaystyle I}$  is a maximal ideal, while ${\displaystyle R/I}$  is an integral domain if and only if ${\displaystyle I}$  is a prime ideal. A number of similar statements relate properties of the ideal ${\displaystyle I}$  to properties of the quotient ring ${\displaystyle R/I}$ .

The Chinese remainder theorem states that, if the ideal ${\displaystyle I}$  is the intersection (or equivalently, the product) of pairwise coprime ideals ${\displaystyle I_{1},\ldots ,I_{k}}$ , then the quotient ring ${\displaystyle R/I}$  is isomorphic to the product of the quotient rings ${\displaystyle R/I_{n},\;n=1,\ldots ,k}$ .

## For algebras over a ring

An associative algebra ${\displaystyle A}$  over a commutative ring ${\displaystyle R}$  is a ring itself. If ${\displaystyle I}$  is an ideal in ${\displaystyle A}$  (closed under ${\displaystyle R}$ -multiplication), then ${\displaystyle A/I}$  inherits the structure of an algebra over ${\displaystyle R}$  and is the quotient algebra.