Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring[1] or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.[2][3] One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I, whose elements are the cosets of I in R subject to special + and operations.

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 ab 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 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 well-defined. Compare coset and quotient group.) The zero-element of R/I is (0 + I) = I, and the multiplicative identity is (1 + 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.


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]/(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 alternative 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, an alternative complex number 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 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 alternatives

Suppose X and Y are two, non-commuting, indeterminates and form the free algebra Then Hamilton’s quaternions of 1843 can be cast as

If Y2  1 is substituted for Y2 + 1, then one obtains the ring of split-quaternions. Substituting minus for plus in both the quadratic binomials also results in split-quaternions. The anti-commutative property YX = −XY implies that XY has for its square

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

The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminates RX,Y,Z⟩ and constructing appropriate ideals.


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 : RS whose kernel contains I, then there exists precisely one ring homomorphism g : R/IS 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 : RS 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.

In commutative algebra and algebraic geometry, the following statement is often used: If R ≠ {0} is a commutative ring and I is a maximal ideal, then the quotient ring R/I is a field; if I is only a prime ideal, then R/I is only an integral domain. 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/Ip, p = 1,...,k.

See also


  1. Jacobson, Nathan (1984). Structure of Rings (revised ed.). American Mathematical Soc. ISBN 0-821-87470-5.
  2. Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
  3. Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

Further references

External links

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