Quotient ring

Short description: Reduction of a ring by one of its ideals

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 ⋅ operations. (Only the fraction slash "/" is used in quotient ring notation, not a horizontal fraction bar.)

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 [math]\displaystyle{ R }[/math] and a two-sided ideal [math]\displaystyle{ I }[/math] in [math]\displaystyle{ R }[/math], we may define an equivalence relation [math]\displaystyle{ \sim }[/math] on [math]\displaystyle{ R }[/math] as follows:

[math]\displaystyle{ a\sim b }[/math] if and only if [math]\displaystyle{ a-b }[/math] is in [math]\displaystyle{ I }[/math].

Using the ideal properties, it is not difficult to check that [math]\displaystyle{ \sim }[/math] is a congruence relation. In case [math]\displaystyle{ a\sim b }[/math], we say that [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are congruent modulo [math]\displaystyle{ I }[/math]. The equivalence class of the element [math]\displaystyle{ a }[/math] in [math]\displaystyle{ R }[/math] is given by

[math]\displaystyle{ [a]=a+I := \{a+r: r\in I\} }[/math].

This equivalence class is also sometimes written as [math]\displaystyle{ a \bmod I }[/math] and called the "residue class of [math]\displaystyle{ a }[/math] modulo [math]\displaystyle{ I }[/math]".

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

• [math]\displaystyle{ (a+I)+(b+I)=(a+b)+I }[/math];
• [math]\displaystyle{ (a+I)(b+I)=(ab)+I }[/math].

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

The map [math]\displaystyle{ p }[/math] from [math]\displaystyle{ R }[/math] to [math]\displaystyle{ R/I }[/math] defined by [math]\displaystyle{ p(a)=a+I }[/math] 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 R, and R / R is the zero ring {0}, since, by our definition, for any r in R, we have that [r] = r + R := {r + b : b ∈ R}, 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 ≠ R, then R / I is not the zero ring.
• Consider the ring of integers Z and the ideal of even numbers, denoted by 2Z. Then the quotient ring Z / 2Z has only two elements, the coset 0+2Z consisting of the even numbers and the coset 1+2Z consisting of the odd numbers; applying the definition, [z] = z + 2Z := {z + 2y: 2y ∈ 2Z}, where 2Z is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, F₂. 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 Z / nZ (which has n elements).
• Now consider the ring of polynomials in the variable X with real coefficients, R[X], and the ideal I = (X² + 1) consisting of all multiples of the polynomial X² + 1. The quotient ring R[X] / (X² + 1) is naturally isomorphic to the field of complex numbers C, with the class [X] playing the role of the imaginary unit i. The reason is that we "forced" X² + 1 = 0, i.e. X² = −1, which is the defining property of i. Since any integer exponent of i must be either ±i or ±1, that means all possible polynomials essentially simplify to the form a + bi. (To clarify, the quotient ring R[X] / (X² + 1) is actually naturally isomorphic to the field of all linear polynomials aX+b, a,b ∈ R,where the operations are performed mod X²+1. In return, we have X² = -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 F₃ = Z / 3Z with three elements. The polynomial f(X) = X² + 1 is irreducible over F₃ (since it has no root), and we can construct the quotient ring F₃[X] / (f). This is a field with 3² = 9 elements, denoted by F₉. 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) | x² = y³ } as a subset of the real plane R². The ring of real-valued polynomial functions defined on V can be identified with the quotient ring R[X,Y] / (X² − Y³), and this is the coordinate ring of V. The variety V is now investigated by studying its coordinate ring.
• Suppose M is a C^∞-manifold, and p is a point of M. Consider the ring R = C^∞(M) of all C^∞-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 C^∞-functions on M at p.
• Consider the ring F of finite elements of a hyperreal field *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 *R, together with 0, is an ideal in F, and the quotient ring F / I is isomorphic to the real numbers 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 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 R[X] / (X), R[X] / (X + 1), and R[X] / (X − 1) are all isomorphic to R and gain little interest at first. But note that R[X] / (X²) 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 X². 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] / (X² − 1) does split into R[X] / (X + 1) and R[X] / (X − 1), so this ring is often viewed as the direct sum R ⊕ R. Nevertheless, a variation on complex numbers z = x + y j is suggested by j as a root of X² − 1, compared to i as root of X² + 1 = 0. This plane of split-complex numbers normalizes the direct sum R ⊕ 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 R⟨X, Y⟩. Then Hamilton’s quaternions of 1843 can be cast as

[math]\displaystyle{ \mathbf{R} \langle X, Y \rangle / ( X^2 + 1, Y^2 + 1, XY + YX ) . }[/math]

If Y² − 1 is substituted for Y² + 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 R⟨X, Y, Z⟩ 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 → S whose kernel contains I, there exists precisely one ring homomorphism g : R / I → 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 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 f : R → S induces a ring isomorphism between the quotient ring R / ker(f) and the image 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 ↦ (a + I) + M / I.

The following facts prove useful in commutative algebra and algebraic geometry: for R ≠ {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 R / I.

The Chinese remainder theorem states that, if the ideal I is the intersection (or equivalently, the product) of pairwise coprime ideals I₁, ..., I_k, then the quotient ring R / I is isomorphic to the product of the quotient rings R / I_n, n = 1, ..., 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. 21–3. 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", pages 47 to 51.

External links

• Hazewinkel, Michiel, ed. (2001), "Quotient ring", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/q076920 
• Ideals and factor rings from John Beachy's Abstract Algebra Online