Quotient ring

In, a branch of , a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the s of and the s of. It is a specific example of a, as viewed from the general setting of. One starts with a R and a  I in R, and constructs a new ring, the quotient ring R / I, whose elements are the  of I in R subject to special + and ⋅ operations.

Quotient rings are distinct from the so-called 'quotient field', or, of an as well as from the more general 'rings of quotients' obtained by.

Formal quotient ring construction
Given a ring R and a two-sided ideal I in R, we may define an ~ on R as follows:
 * a ~ b a − b is in I.

Using the ideal properties, it is not difficult to check that ~ is a. In case a ~ b, we say that a and b are congruent  I. The of the element a in R is given by


 * [a] = a + I := { a + r : r in 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 . Compare and .) The zero-element of R / I is (0 + I) = I, and the multiplicative identity is (1 + I).
 * (a + I) + (b + I) = (a + b) + I;
 * (a + I)(b + I) = (a b) + I.

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

Examples

 * The quotient ring R / {0}}{{null} is to R, and R / R is the  {0}, since, by our definition, for any r in R, we have that [r] = r + {0} := {r + b : b ∈ {0}}|undefined (where {0} is the zero ring), which is isomorphic to 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 s Z and the ideal of s, denoted by 2Z. Then the quotient ring Z / 2Z has only two elements, zero for the even numbers and one for 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 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).  is essentially arithmetic in the quotient ring Z / nZ (which has n elements).
 * Now consider the ring R[X] of s in the variable X with coefficients, and the ideal I = (X + 1) consisting of all multiples of the polynomial X + 1. The quotient ring R[X] / (X + 1) is naturally isomorphic to the field of s C, with the class [X] playing the role of the  i. The reason is that we "forced" X + 1 = 0, i.e. X = −1, which is the defining property of i.
 * Generalizing the previous example, quotient rings are often used to construct s. Suppose K is some and f is an  in K[X]. Then L = K[X] / (f) is a field whose  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 s of are important examples of quotient rings in . As a simple case, consider the real variety V = {(x, y) 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∞-, 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 U of p (where U may depend on f). Then the quotient ring R / I is the ring of s of C∞-functions on M at p.
 * Consider the ring F of finite elements of a *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  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 s), see.

Alternative 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 plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of R[X] by X. This alternative complex plane arises as a whenever the algebra contains a  and a.

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 R ⊕ R. Nevertheless, an alternative complex number 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 s 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 may be compared to the  of the.

Quaternions and alternatives
Suppose X and Y are two, non-commuting, s and form the R$⟨X, Y⟩$. Then Hamilton’s s of 1843 can be cast as
 * $$\mathbf{R} \langle X, Y \rangle / ( X^2 + 1, Y^2 + 1, XY + YX ) .$$

If Y − 1 is substituted for Y + 1, then one obtains the ring of s. Substituting minus for plus in both the quadratic binomials also results in split-quaternions. The YX = −XY implies that XY has as its square


 * (XY)(XY) = X(YX)Y = −X(XY)Y = −XXYY = −1.

The three types of s 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, then so is R / I; the converse however is not true in general.

The natural quotient map p has I as its ; 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 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 between the quotient ring R / ker(f) and the image im(f). (See also: .)

The ideals of R and R / I are closely related: the natural quotient map provides a 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.

In and, the following statement is often used: If R ≠ {0} is a  ring and I is a , then the quotient ring R / I is a ; if I is only a , then R / I is only an. A number of similar statements relate properties of the ideal I to properties of the quotient ring R / I.

The states that, if the ideal I is the intersection (or equivalently, the product) of pairwise  ideals I1, ..., Ik, then the quotient ring R / I is isomorphic to the  of the quotient rings R / Ip, p = 1, ..., k.

For algebras over a ring
An A over a  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.