Quotient ring

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.

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.

• 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 ${\displaystyle -n<x<n}$ 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.

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.

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.

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 ${\displaystyle R/M}$ 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 ${\displaystyle R/I_{n},\;n=1,\ldots ,k}$.

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.

• Associated graded ring
• Residue field
• Goldie's theorem
• Quotient module

• 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.
• Joseph Rotman (1998). Galois Theory (2nd ed.). Springer. pp. 21–23. ISBN 0-387-98541-7.
• 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.

• "Quotient ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Ideals and factor rings from John Beachy's Abstract Algebra Online