Algebraic structure → Ring theory Ring theory |
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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 group in group theory and to the quotient space in linear algebra.^{[2]}^{[3]} 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.

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 2**Z**, 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

- (
*a*+*I*) + (*b*+ I*)*= (*a*+*b*) +*I*; - (
*a*+*I*)(*b*+*I*) = (*ab*) +*I*.

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of *R* / *I* is 0 = (0 + *I*) = *I*, and the multiplicative identity is 1 = (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

- 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 2**Z**. Then the quotient ring**Z**/ 2**Z**has only two elements, the coset 0 + 2**Z**consisting of the even numbers and the coset 1 + 2**Z**consisting of the odd numbers; applying the definition, [*z*] =*z*+ 2**Z**= {*z*+ 2*y*: 2*y*∈ 2**Z**}, where 2**Z**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**/*n***Z**(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
**F**_{3}=**Z**/ 3**Z**with three elements. The polynomial*f*(*X*) =*X*^{2}+ 1 is irreducible over**F**_{3}(since it has no root), and we can construct the quotient ring**F**_{3}[*X*] / (*f*). This is a field with 3^{2}= 9 elements, denoted by**F**_{9}. 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**R**^{2}. The ring of real-valued polynomial functions defined on*V*can be identified with the quotient ring**R**[*X*,*Y*] / (*X*^{2}−*Y*^{3}), 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.

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*^{2}) 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*^{2}. 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*^{2} − 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*^{2} − 1, compared to i as root of *X*^{2} + 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.

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*^{2} − 1 is substituted for *Y*^{2} + 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.

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 *I*_{1}, ..., *I _{k}*, then the quotient 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**.