Quotient ring

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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] It is a specific example of a quotient, as viewed from the general setting of universal algebra. 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.

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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 and a two-sided ideal in , we may define an equivalence relation on as follows:

if and only if is in .

Using the ideal properties, it is not difficult to check that is a congruence relation. In case , we say that and are congruent modulo . The equivalence class of the element in is given by

.

This equivalence class is also sometimes written as and called the "residue class of modulo ".

The set of all such equivalence classes is denoted by ; it becomes a ring, the factor ring or quotient ring of modulo , if one defines

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of is , and the multiplicative identity is .

The map from to defined by is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism .

Examples

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 RR. 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 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 alternatives

Suppose X and Y are two, non-commuting, indeterminates and form the free algebra RX, Y. 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 as 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.

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 : RS whose kernel contains I, 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.

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.

See also

Notes

  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

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