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. (Only the fraction slash "/" is used in quotient ring notation, not a horizontal fraction bar.)

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 ${\displaystyle R}$ and a two-sided ideal ${\displaystyle I}$ in ${\displaystyle R}$, we may define an equivalence relation ${\displaystyle \sim }$ on ${\displaystyle R}$ as follows:

${\displaystyle a\sim b}$ if and only if ${\displaystyle a-b}$ is in ${\displaystyle I}$.

Using the ideal properties, it is not difficult to check that ${\displaystyle \sim }$ is a congruence relation. In case ${\displaystyle a\sim b}$, we say that ${\displaystyle a}$ and ${\displaystyle b}$ are congruent modulo ${\displaystyle I}$. The equivalence class of the element ${\displaystyle a}$ in ${\displaystyle R}$ is given by

${\displaystyle [a]=a+I:=\{a+r:r\in I\))$.

This equivalence class is also sometimes written as ${\displaystyle a{\bmod {I))}$ and called the "residue class of ${\displaystyle a}$ modulo ${\displaystyle I}$".

The set of all such equivalence classes is denoted by ${\displaystyle R/I}$; it becomes a ring, the factor ring or quotient ring of ${\displaystyle R}$ modulo ${\displaystyle I}$, if one defines

• ${\displaystyle (a+I)+(b+I)=(a+b)+I}$;
• ${\displaystyle (a+I)(b+I)=(ab)+I}$.

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

The map ${\displaystyle p}$ from ${\displaystyle R}$ to ${\displaystyle R/I}$ defined by ${\displaystyle p(a)=a+I}$ is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism.

Examples

• The quotient ring ${\displaystyle R/\!\left\{0\right\))$ is naturally isomorphic to ${\displaystyle R}$, and ${\displaystyle R/\!R}$ is the zero ring ${\displaystyle \left\{0\right\))$, since, by our definition, for any ${\displaystyle r\in R}$, we have that ${\displaystyle \left[r\right]=r+R=\left\{r+b:b\in R\right\))$, which equals ${\displaystyle R}$ itself. This fits with the rule of thumb that the larger the ideal ${\displaystyle I}$, the smaller the quotient ring ${\displaystyle R/I}$. If ${\displaystyle I}$ is a proper ideal of ${\displaystyle R}$, i.e., ${\displaystyle I\neq R}$, then ${\displaystyle R/I}$ is not the zero ring.
• Consider the ring of integers ${\displaystyle \mathbb {Z} }$ and the ideal of even numbers, denoted by ${\displaystyle 2\mathbb {Z} }$. Then the quotient ring ${\displaystyle 2\mathbb {Z} }$ has only two elements, the coset ${\displaystyle 0+2\mathbb {Z} }$ consisting of the even numbers and the coset ${\displaystyle 1+2\mathbb {Z} }$ consisting of the odd numbers; applying the definition, ${\displaystyle \left[z\right]=z+2\mathbb {Z} =\left\{z+2y:2y\in 2\mathbb {Z} \right\))$, where ${\displaystyle 2\mathbb {Z} }$ is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, ${\displaystyle \mathbb {F} _{2))$. Intuitively: if you think of all the even numbers as ${\displaystyle 0}$, then every integer is either ${\displaystyle 0}$ (if it is even) or ${\displaystyle 1}$ (if it is odd and therefore differs from an even number by ${\displaystyle 1}$). Modular arithmetic is essentially arithmetic in the quotient ring ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ (which has ${\displaystyle n}$ elements).
• Now consider the ring of polynomials in the variable ${\displaystyle X}$ with real coefficients, ${\displaystyle \mathbb {R} \left[X\right]}$, and the ideal ${\displaystyle I=\left(X^{2}+1\right)}$ consisting of all multiples of the polynomial ${\displaystyle X^{2}+1}$. The quotient ring ${\displaystyle \mathbb {R} \left[X\right]/\left(X^{2}+1\right)}$ is naturally isomorphic to the field of complex numbers ${\displaystyle \mathbb {C} }$, with the class ${\displaystyle [X]}$ playing the role of the imaginary unit ${\displaystyle i}$. The reason is that we "forced" ${\displaystyle X^{2}+1=0}$, i.e. ${\displaystyle X^{2}=-1}$, which is the defining property of ${\displaystyle i}$. Since any integer exponent of ${\displaystyle i}$ must be either ${\displaystyle \pm i}$ or ${\displaystyle \pm 1}$, that means all possible polynomials essentially simplify to the form ${\displaystyle a+bi}$. (To clarify, the quotient ring ${\displaystyle \mathbb {R} \left[X\right]/\left(X^{2}+1\right)}$ is actually naturally isomorphic to the field of all linear polynomials ${\displaystyle aX+b}$, ${\displaystyle a,b\in \mathbb {R} }$, where the operations are performed ${\displaystyle {\textrm {mod))\left(X^{2}+1\right)}$. In return, we have ${\displaystyle X^{2}=-1}$, and this is matching ${\displaystyle X}$ 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^∞-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.

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 dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of R[X] by X². 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] / (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, a variation on complex numbers 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 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

${\displaystyle \mathbf {R} \langle X,Y\rangle /(X^{2}+1,Y^{2}+1,XY+YX).}$

If Y² − 1 is substituted for Y² + 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.

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.

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, ..., I_k, then the quotient ring R / I is isomorphic to the product of the quotient rings R / I_n, 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.