Quotient ring

In ring theory, a branch of abstract algebra, a **quotient ring**, also known as **factor ring**, **difference ring** or **residue class ring**, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra. 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.

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 *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* 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

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

## Examples

* 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* in *R*, we have that [*r*] = *r* + {0} := {*r* + *b* : *b* ∈ {0}} (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 integers **Z** and the ideal of even numbers, denoted by 2**Z**. (Odd numbers form a coset.) Then the quotient ring **Z** / 2**Z** has only two elements (one for each coset), zero for the even numbers and one for 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 **R**[*X*] of polynomials in the variable *X* with real coefficients, 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*.

* 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 32 = 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.

### 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*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 alternative 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, an alternative complex number *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.

### Quaternions and alternatives

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
,
X
Y
Y
X
)
.

{\displaystyle \mathbf{R} \langle X, Y \rangle / ( 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. 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 **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 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 *I*1, ..., *Ik*, then the quotient ring *R* / *I* is isomorphic to the product of the quotient rings *R* / *Ip*, *p* = 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**.