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 Template:Tmath, a new ring, the quotient ring Template:Tmath, 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 "Template:Tmath".)

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 Template:Tmath, we may define an equivalence relation $\sim$ on $R$ as follows:

$a\sim b$ if and only if $a-b$ is in Template:Tmath.

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

This equivalence class is also sometimes written as $amodI$ and called the "residue class of $a$ modulo $I$".

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

• Template:Tmath;
• Template:Tmath.

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

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 $R/\{0\}$ is naturally isomorphic to Template:Tmath, and $R/R$ is the zero ring Template:Tmath, since, by our definition, for any Template:Tmath, we have that Template:Tmath, which equals $R$ itself. This fits with the rule of thumb that the larger the ideal Template:Tmath, the smaller the quotient ring Template:Tmath. If $I$ is a proper ideal of Template:Tmath, i.e., Template:Tmath, then $R/I$ is not the zero ring.
• Consider the ring of integers $\mathbb{Z}$ and the ideal of even numbers, denoted by Template:Tmath. 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, Template:Tmath, where $2\mathbb{Z}$ is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, Template:Tmath. Intuitively: if you think of all the even numbers as Template:Tmath, then every integer is either $0$ (if it is even) or $1$ (if it is odd and therefore differs from an even number by Template:Tmath). 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, Template:Tmath, and the ideal $I=(X^2+1)$ consisting of all multiples of the polynomial Template:Tmath. The quotient ring $ℝ[X]/(X^2+1)$ is naturally isomorphic to the field of complex numbers Template:Tmath, with the class $[X]$ playing the role of the imaginary unit Template:Tmath. The reason is that we "forced" Template:Tmath, i.e. Template:Tmath, which is the defining property of Template:Tmath. Since any integer exponent of $i$ must be either $\pm i$ or Template:Tmath, that means all possible polynomials essentially simplify to the form Template:Tmath. (To clarify, the quotient ring Template:Tmath is actually naturally isomorphic to the field of all linear polynomials Template:Tmath, where the operations are performed modulo Template:Tmath. In return, we have Template:Tmath, 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 Template:Tmath. Then $L=K[X]/(f)$ is a field whose minimal polynomial over $K$ is Template:Tmath, which contains $K$ as well as an element Template:Tmath.
• 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 $f(X)=X{i}^2+1$ is irreducible over $F_3$ (since it has no root), and we can construct the quotient ring Template:Tmath. This is a field with $3^2=9$ elements, denoted by Template:Tmath. 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 $V=\{(x,y)|x^2=y^3\}$ as a subset of the real plane Template:Tmath. The ring of real-valued polynomial functions defined on $V$ can be identified with the quotient ring Template:Tmath, and this is the coordinate ring of Template:Tmath. The variety $V$ is now investigated by studying its coordinate ring.
• Suppose $M$ is a ${ℂ}^{\mathrm{\infty }}$-manifold, and $p$ is a point of Template:Tmath. Consider the ring $R={ℂ}^{\mathrm{\infty }}(M)$ of all ${ℂ}^{\mathrm{\infty }}$-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 Template:Tmath). Then the quotient ring $R/I$ is the ring of germs of ${ℂ}^{\mathrm{\infty }}$-functions on $M$ at Template:Tmath.
• Consider the ring $F$ of finite elements of a hyperreal field Template:Tmath. 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 Template:Tmath, together with Template:Tmath, is an ideal in Template:Tmath, and the quotient ring $F/I$ is isomorphic to the real numbers Template:Tmath. The isomorphism is induced by associating to every element $x$ of $F$ the standard part of Template:Tmath, i.e. the unique real number that differs from $x$ by an infinitesimal. In fact, one obtains the same result, namely Template:Tmath, 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 Template:Tmath, Template:Tmath, and $ℝ[X]/(X-1)$ are all isomorphic to $ℝ$ and gain little interest at first. But note that $ℝ[X]/(X^2)$ is called the dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of $ℝ[X]$ by Template:Tmath. This variation of a complex plane arises as a subalgebra whenever the algebra contains a real line and a nilpotent.

Furthermore, the ring quotient $ℝ[X]/(X^2-1)$ does split into $ℝ[X]/(X+1)$ and Template:Tmath, so this ring is often viewed as the direct sum Template:Tmath. Nevertheless, a variation on complex numbers $z=x+yj$ is suggested by $j$ as a root of Template:Tmath, compared to $i$ as root of Template:Tmath. This plane of split-complex numbers normalizes the direct sum $ℝ\oplus ℝ$ by providing a basis $\{1,j\}$ 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 Template:Tmath. Then Hamilton's quaternions of 1843 can be cast as: $$ℝ⟨X,Y⟩/(X^2+1,Y^2+1,XY+YX)$$

If $Y^2-1$ is substituted for Template:Tmath, 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 $ℝ⟨X,Y,Z⟩$ and constructing appropriate ideals.

Properties

Clearly, if $R$ is a commutative ring, then so is Template:Tmath; 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 Template:Tmath. More precisely, given a two-sided ideal $I$ in $R$ and a ring homomorphism $f:R\to S$ whose kernel contains Template:Tmath, 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 Template:Tmath. 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 Template:Tmath. (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 Template:Tmath, and we write $M/I$ for the corresponding ideal in $R/I$ (i.e. Template:Tmath), the quotient rings $R/M$ and $(R/I)/(M/I)$ are naturally isomorphic via the (well-defined) mapping Template:Tmath.

The following facts prove useful in commutative algebra and algebraic geometry: for $R\ne \{0\}$ 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 Template:Tmath.

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

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.
• Template:Cite book
• 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

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