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 ⋅ 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 ~ on R as follows:
 * a ~ b if and only if a − b is in I.

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

This equivalence class is also sometimes written as a mod 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 I, if one defines (Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of R / I is $\overline{0}$ = (0 + I) = I, and the multiplicative identity is $\overline{1}$ = (1 + I).
 * (a + I) + (b + I) = (a + b) + I;
 * (a + I)(b + I) = (ab) + 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 / $\{0\}$ is naturally isomorphic to R, and R / R is the zero ring $\{0\}$, since, by our definition, for any r ∈ 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, 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 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 = (X$2$ + 1) consisting of all multiples of the polynomial X$2$ + 1. The quotient ring R[X] / (X$2$ + 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" 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 ±i or ±1, that means all possible polynomials essentially simplify to the form a + bi. (To clarify, the quotient ring R[X] / (X$2$ + 1) is actually naturally isomorphic to the field of all linear polynomials aX + b, a, b ∈ R, where the operations are performed mod (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 F3 = Z / 3Z with three elements. The polynomial f(X) = X + 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 = $\{(x, y) | x^{2} = y^{3}\}$ 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] / (X − 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] / (X) 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 X. 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 − 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 X − 1, compared to i as root of X + 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
 * R$⟨X, Y⟩$ / (X$2$ + 1, Y$2$ + 1, XY + YX).

If Y − 1 is substituted for Y + 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, ..., Ik, then the quotient ring R / I is isomorphic to the product of the quotient rings R / In, 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.

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.
 * 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.
 * 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.