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 $R$ and a two-sided ideal $I$ in ⁠$R$⁠, a new ring, the quotient ring ⁠$R\ /\ I$⁠, is constructed, whose elements are the cosets of $I$ in $R$ subject to special $+$ and $\cdot$ 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 $R$ and a two-sided ideal $I$ in ⁠$R$⁠, we may define an equivalence relation $\sim$ on $R$ as follows:

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

Using the ideal properties, it is not difficult to check that $\sim$ is a congruence relation. In case ⁠$a\sim b$⁠, 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 ⁠$2\mathbb {Z}$⁠, the even integers). The equivalence class of the element $a$ in $R$ is given by: $$\left[a\right]=a+I:=\left\lbrace a+r:r\in I\right\rbrace$$

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

The set of all such equivalence classes is denoted by ⁠$R\ /\ I$⁠; it becomes a ring, the factor ring or quotient ring of $R$ modulo ⁠${1}$⁠, if one defines

• ⁠$(a+I)+(b+I)=(a+b)+I$⁠;
• ⁠$(a+I)(b+I)=(ab)+I$⁠.

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

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

Examples

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

Variations of complex planes

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

Furthermore, the ring quotient $\mathbb {R} [X]/(X^{2}-1)$ does split into $\mathbb {R} [X]/(X+1)$ and ⁠$\mathbb {R} [X]/(X-1)$⁠, so this ring is often viewed as the direct sum ⁠$\mathbb {R} \oplus \mathbb {R}$⁠. Nevertheless, a variation on complex numbers $z=x+yj$ is suggested by $j$ as a root of ⁠$X^{2}-1=0$⁠, compared to $i$ as root of ⁠$X^{2}+1=0$⁠. This plane of split-complex numbers normalizes the direct sum $\mathbb {R} \oplus \mathbb {R}$ by providing a basis $\left\lbrace 1,j\right\rbrace$ 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 $X$ and $Y$ are two non-commuting indeterminates and form the free algebra ⁠$\mathbb {R} \langle X,Y\rangle$⁠. Then Hamilton's quaternions of 1843 can be cast as: $$\mathbb {R} \langle X,Y\rangle /(X^{2}+1,\,Y^{2}+1,\,XY+YX)$$

If $Y^{2}-1$ is substituted for ⁠$Y^{2}+1$⁠, then one obtains the ring of split-quaternions. 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 biquaternions can also be written as quotients by use of the free algebra with three indeterminates $\mathbb {R} \langle X,Y,Z\rangle$ and constructing appropriate ideals.

Properties

Clearly, if $R$ is a commutative ring, then so is ⁠$R\ /\ I$⁠; 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 $R\ /\ I$ are essentially the same as the ring homomorphisms defined on $R$ that vanish (i.e. are zero) on ⁠$I$⁠. More precisely, given a two-sided ideal $I$ in $R$ and a ring homomorphism $f:R\to S$ whose kernel contains ⁠$I$⁠, there exists precisely one ring homomorphism $g:R\ /\ I\to S$ 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 ⁠$1R$⁠. 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 $f:R\to S$ induces a ring isomorphism between the quotient ring $R\ /\ \ker(f)$ and the image ⁠$\mathrm {im} (f)$⁠. (See also: Fundamental theorem on homomorphisms.)

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

The following facts prove useful in commutative algebra and algebraic geometry: for $R\neq \lbrace 0\rbrace$ commutative, $R\ /\ I$ 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 ⁠$R\ /\ I$⁠.

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

For algebras over a ring

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

See also

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

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