Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. 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 "⁠${\displaystyle /}$⁠".)

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

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.

See also

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

Notes

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.
• 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.

External links

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