Quotient ring

 ( 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.

(2025). 9780471433347, John Wiley & Sons. ISBN 9780471433347

Serge Lang (2025). 038795385X, Springer. 038795385X

It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring $R$ and a two-sided ideal $I$ in , a new ring, the quotient ring , is constructed, whose elements are the cosets of $I$ in $R$ subject to special $+$ and $\backslash cdot$ operations. (Quotient ring notation almost always uses a fraction slash ""; stacking the ring over the ideal using a horizontal line as a separator is uncommon and generally avoided.)

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.

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Formal quotient ring construction Given a ring $R$ and a two-sided ideal $I$ in , we may define an equivalence relation $\backslash sim$ on $R$ as follows:

$a\; \backslash sim\; b$ if and only if $a\; -\; b$ is in .

Using the ideal properties, it is not difficult to check that $\backslash sim$ is a congruence relation. In case , we say that $a$ and $b$ are congruent modulo $I$ (for example, $1$ and $3$ are congruent modulo $2$ as their difference is an element of the ideal , the even integers). The equivalence class of the element $a$ in $R$ is given by: $$\backslash left\; =\; \backslash overline\{a\}\; =\; a\; +\; I\; :=\; \backslash left\backslash lbrace\; a\; +\; r\; :\; r\; \backslash in\; I\; \backslash right\backslash rbrace$$This equivalence class is also sometimes written as $a\; \backslash bmod\; I$ and called the "residue class of $a$ modulo $I$".

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

• ;
• .

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of $R\backslash \; /\backslash \; I$ is , and the multiplicative identity is .

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

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Examples

• The quotient ring $R\backslash \; /\backslash \; \backslash lbrace\; 0\; \backslash rbrace$ is naturally isomorphic to , and $R\; /\; R$ is the zero ring , since, by our definition, for any , we have that , which equals $R$ itself. This fits with the rule of thumb that the larger the ideal , the smaller the quotient ring . If $I$ is a proper ideal of , i.e., , then $R\; /\; I$ is not the zero ring.
• Consider the ring of $\backslash mathbb\{Z\}$ and the ideal of , denoted by . Then the quotient ring $\backslash mathbb\{Z\}\; /\; 2\; \backslash mathbb\{Z\}$ has only two elements, the coset $0\; +\; 2\; \backslash mathbb\{Z\}$ consisting of the even numbers and the coset $1\; +\; 2\; \backslash mathbb\{Z\}$ consisting of the odd numbers; applying the definition, , where $2\; \backslash mathbb\{Z\}$ is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, . Intuitively: if you think of all the even numbers as , then every integer is either $0$ (if it is even) or $1$ (if it is odd and therefore differs from an even number by ). Modular arithmetic is essentially arithmetic in the quotient ring $\backslash mathbb\{Z\}\; /\; n\; \backslash mathbb\{Z\}$ (which has $n$ elements).
• Now consider the ring of polynomials in the variable $X$ with real number , , and the ideal $I\; =\; \backslash left(\; X^2\; +\; 1\; \backslash right)$ consisting of all multiples of the polynomial . The quotient ring $\backslash mathbb\{R\}\; X\backslash \; /\backslash \; (\; X^2\; +\; 1\; )$ is naturally isomorphic to the field of , with the class $X$ playing the role of the imaginary unit . The reason is that we "forced" , i.e. , which is the defining property of . Since any integer exponent of $i$ must be either $\backslash pm\; i$ or , that means all possible polynomials essentially simplify to the form . (To clarify, the quotient ring is actually naturally isomorphic to the field of all linear polynomials , where the operations are performed modulo . In return, we have , and this is matching $X$ to the imaginary unit in the isomorphic field of complex numbers.)
• Generalizing the previous example, quotient rings are often used to construct . Suppose $K$ is some field and $f$ is an irreducible polynomial in . Then $L\; =\; KX\backslash \; /\backslash \; (f)$ is a field whose minimal polynomial over $K$ is , which contains $K$ as well as an element .
• One important instance of the previous example is the construction of the finite fields. Consider for instance the field $F\_3\; =\; \backslash mathbb\{Z\}\; /\; 3\backslash mathbb\{Z\}$ with three elements. The polynomial $f(X)\; =\; X^2\; +1$ is irreducible over $F\_3$ (since it has no root), and we can construct the quotient ring . This is a field with $3^2\; =\; 9$ elements, denoted by . The other finite fields can be constructed in a similar fashion.
• The of algebraic varieties are important examples of quotient rings in algebraic geometry. As a simple case, consider the real variety $V\; =\; \backslash left\backslash lbrace\; (x,y)\; |\; x^2\; =\; y^3\; \backslash right\backslash rbrace$ as a subset of the real plane . The ring of real-valued polynomial functions defined on $V$ can be identified with the quotient ring , and this is the coordinate ring of . The variety $V$ is now investigated by studying its coordinate ring.
• Suppose $M$ is a $\backslash mathbb\{C\}^\{\backslash infty\}$-manifold, and $p$ is a point of . Consider the ring $R\; =\; \backslash mathbb\{C\}^\{\backslash infty\}(M)$ of all $\backslash mathbb\{C\}^\{\backslash infty\}$-functions defined on $M$ and let $I$ be the ideal in $R$ consisting of those functions $f$ which are identically zero in some neighborhood $U$ of $p$ (where $U$ may depend on ). Then the quotient ring $R\backslash \; /\backslash \; I$ is the ring of germs of $\backslash mathbb\{C\}^\{\backslash infty\}$-functions on $M$ at .
• Consider the ring $F$ of finite elements of a hyperreal number . 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 exists. The set $I$ of all infinitesimal numbers in , together with , is an ideal in , and the quotient ring $F\backslash \; /\backslash \; I$ is isomorphic to the real numbers . The isomorphism is induced by associating to every element $x$ of $F$ the standard part of , i.e. the unique real number that differs from $x$ by an infinitesimal. In fact, one obtains the same result, namely , if one starts with the ring $F$ of finite hyperrationals (i.e. ratio of a pair of ), see construction of the real numbers.

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

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

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

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

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

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

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Properties Clearly, if $R$ is a commutative ring, then so is ; the converse, however, is not true in general.

The natural quotient map $p$ has $I$ 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 $R\backslash \; /\backslash \; I$ are essentially the same as the ring homomorphisms defined on $R$ that vanish (i.e. are zero) on . More precisely, given a two-sided ideal $I$ in $R$ and a ring homomorphism $f\; :\; R\; \backslash to\; S$ whose kernel contains , there exists precisely one ring homomorphism $g\; :\; R\backslash \; /\backslash \; I\; \backslash to\; S$ 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 . 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 $f\; :\; R\; \backslash to\; S$ induces a ring isomorphism between the quotient ring $R\backslash \; /\backslash \; \backslash ker\; (f)$ and the image . (See also: Fundamental theorem on homomorphisms.)

The ideals of $R$ and $R\backslash \; /\backslash \; I$ 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 $R\backslash \; /\backslash \; I$ (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 , and we write $M\backslash \; /\backslash \; I$ for the corresponding ideal in $R\backslash \; /\backslash \; I$ (i.e. ), the quotient rings $R\backslash \; /\backslash \; M$ and $(R\; /\; I)\backslash \; /\backslash \; (M\; /\; I)$ are naturally isomorphic via the (well-defined) mapping .

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

The Chinese remainder theorem states that, if the ideal $I$ is the intersection (or equivalently, the product) of pairwise coprime ideals , then the quotient ring $R\backslash \; /\backslash \; I$ is isomorphic to the product of the quotient rings .

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

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See also

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

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Notes

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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). 9780387985411, Springer. ISBN 9780387985411
• 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.

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External links

• Ideals and factor rings from John Beachy's Abstract Algebra Online

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