Quotient ring

In

Quotient rings are distinct from the so-called "quotient field", or

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. 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

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

(Here one has to check that these definitions are

• 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
  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^∞-
  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

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⟩ / (X² + 1, Y² + 1, XY + YX).

If Y² − 1 is substituted for Y² + 1, then one obtains the ring of

(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

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

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

Notes