Inring theory, a branch ofabstract algebra, aquotient ring, also known asfactor ring,difference ring^[1] orresidue class ring, is a construction quite similar to thequotient group ingroup theory and to thequotient space inlinear algebra.^[2][3] It is a specific example of aquotient, as viewed from the general setting ofuniversal algebra. Starting with aring${\displaystyle R}$ and atwo-sided ideal${\displaystyle I}$ in⁠${\displaystyle R}$⁠, a new ring, the quotient ring⁠${\displaystyle R/I}$⁠, is constructed, whose elements are thecosets of${\displaystyle I}$ in${\displaystyle R}$ subject to special${\displaystyle +}$ and${\displaystyle \cdot }$ operations. (Quotient ring notation always uses afraction slash "⁠${\displaystyle /}$⁠".)

Quotient rings are distinct from the so-called "quotient field", orfield of fractions, of anintegral domain as well as from the more general "rings of quotients" obtained bylocalization.

Given a ring${\displaystyle R}$  and a two-sided ideal${\displaystyle I}$  in⁠${\displaystyle R}$ ⁠, we may define anequivalence 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 acongruence relation.In case⁠${\displaystyle a\sim b}$ ⁠, we say that${\displaystyle a}$  and${\displaystyle b}$  arecongruentmodulo${\displaystyle I}$  (for example,${\displaystyle 1}$  and${\displaystyle 3}$  are congruent modulo${\displaystyle 2}$  as their difference is an element of the ideal⁠${\displaystyle 2\mathbb{Z}}$ ⁠, theeven integers). Theequivalence class of the element${\displaystyle a}$  in${\displaystyle R}$  is given by:$${\displaystyle \left[a\right]=a+I:=\left\{a+r:r\in I\right\}}$$ 

This equivalence class is also sometimes written as${\displaystyle amodI}$  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, thefactor ring orquotient 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 arewell-defined. Comparecoset andquotient group.) The zero-element of${\displaystyle R/I}$  is⁠${\displaystyle \overline{0}=(0+I)=I}$ ⁠, and the multiplicative identity is⁠${\displaystyle \overline{1}=(1+I)}$ ⁠.

The map${\displaystyle p}$  from${\displaystyle R}$  to${\displaystyle R/I}$  defined by${\displaystyle p(a)=a+I}$  is asurjectivering homomorphism, sometimes called thenatural quotient map or thecanonical homomorphism.

• The quotient ring${\displaystyle R/\{0\}}$  isnaturally isomorphic to⁠${\displaystyle R}$ ⁠, and${\displaystyle R/R}$  is thezero ring⁠${\displaystyle \{0\}}$ ⁠, 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.,⁠${\displaystyle I\ne R}$ ⁠, then${\displaystyle R/I}$  is not the zero ring.
• Consider the ring ofintegers${\displaystyle \mathbb{Z}}$  and the ideal ofeven numbers, denoted by⁠${\displaystyle 2\mathbb{Z}}$ ⁠. Then the quotient ring${\displaystyle \mathbb{Z}/2\mathbb{Z}}$  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 thefinite field with two elements,⁠${\displaystyle F_2}$ ⁠. 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 thering of polynomials in the variable${\displaystyle X}$  withrealcoefficients,⁠${\displaystyle \mathbb{R}[X]}$ ⁠, and the ideal${\displaystyle I=\left(X^2+1\right)}$  consisting of all multiples of thepolynomial⁠${\displaystyle X^2+1}$ ⁠. The quotient ring${\displaystyle \mathbb{R}[X]/(X^2+1)}$  is naturally isomorphic to the field ofcomplex numbers⁠${\displaystyle \mathbb{C}}$ ⁠, with the class${\displaystyle [X]}$  playing the role of theimaginary 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${\displaystyle \mathbb{R}[X]/(X^2+1)}$ ⁠ is actually naturally isomorphic to the field of all linear polynomials⁠${\displaystyle aX+b;a,b\in \mathbb{R}}$ ⁠, where the operations are performed modulo⁠${\displaystyle X^2+1}$ ⁠. 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 constructfield extensions. Suppose${\displaystyle K}$  is somefield and${\displaystyle f}$  is anirreducible polynomial in⁠${\displaystyle K[X]}$ ⁠. Then${\displaystyle L=K[X]/(f)}$  is a field whoseminimal polynomial over${\displaystyle K}$  is⁠${\displaystyle f}$ ⁠, which contains${\displaystyle K}$  as well as an element⁠${\displaystyle x=X+(f)}$ ⁠.
• One important instance of the previous example is the construction of the finite fields. Consider for instance the field${\displaystyle F_3=\mathbb{Z}/3\mathbb{Z}}$  with three elements. The polynomial${\displaystyle f(X)=X{i}^2+1}$  is irreducible over${\displaystyle F_3}$  (since it has no root), and we can construct the quotient ring⁠${\displaystyle F_3[X]/(f)}$ ⁠. This is a field with${\displaystyle 3^2=9}$  elements, denoted by⁠${\displaystyle F_9}$ ⁠. The other finite fields can be constructed in a similar fashion.
• Thecoordinate rings ofalgebraic varieties are important examples of quotient rings inalgebraic geometry. As a simple case, consider the real variety${\displaystyle V=\left\{(x,y)|x^2=y^3\right\}}$  as a subset of the real plane⁠${\displaystyle {\mathbb{R}}^2}$ ⁠. The ring of real-valued polynomial functions defined on${\displaystyle V}$  can be identified with the quotient ring⁠${\displaystyle \mathbb{R}[X,Y]/(X^2-Y^3)}$ ⁠, and this is the coordinate ring of⁠${\displaystyle V}$ ⁠. The variety${\displaystyle V}$  is now investigated by studying its coordinate ring.
• Suppose${\displaystyle M}$  is a${\displaystyle {\mathbb{C}}^{\mathrm{\infty }}}$ -manifold, and${\displaystyle p}$  is a point of⁠${\displaystyle M}$ ⁠. Consider the ring${\displaystyle R={\mathbb{C}}^{\mathrm{\infty }}(M)}$  of all${\displaystyle {\mathbb{C}}^{\mathrm{\infty }}}$ -functions defined on${\displaystyle M}$  and let${\displaystyle I}$  be the ideal in${\displaystyle R}$  consisting of those functions${\displaystyle f}$  which are identically zero in someneighborhood${\displaystyle U}$  of${\displaystyle p}$  (where${\displaystyle U}$  may depend on⁠${\displaystyle f}$ ⁠). Then the quotient ring${\displaystyle R/I}$  is the ring ofgerms of${\displaystyle {\mathbb{C}}^{\mathrm{\infty }}}$ -functions on${\displaystyle M}$  at⁠${\displaystyle p}$ ⁠.
• Consider the ring${\displaystyle F}$  of finite elements of ahyperreal field⁠${\displaystyle {}^{\ast }\mathbb{R}}$ ⁠. It consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers${\displaystyle x}$  for which a standard integer${\displaystyle n}$  with  exists. The set${\displaystyle I}$  of all infinitesimal numbers in⁠${\displaystyle {}^{\ast }\mathbb{R}}$ ⁠, together with⁠${\displaystyle 0}$ ⁠, is an ideal in⁠${\displaystyle F}$ ⁠, and the quotient ring${\displaystyle F/I}$  is isomorphic to the real numbers⁠${\displaystyle \mathbb{R}}$ ⁠. The isomorphism is induced by associating to every element${\displaystyle x}$  of${\displaystyle F}$  thestandard part of⁠${\displaystyle x}$ ⁠, i.e. the unique real number that differs from${\displaystyle x}$  by an infinitesimal. In fact, one obtains the same result, namely⁠${\displaystyle \mathbb{R}}$ ⁠, if one starts with the ring${\displaystyle F}$  of finite hyperrationals (i.e. ratio of a pair ofhyperintegers), seeconstruction of the real numbers.

The quotients⁠${\displaystyle \mathbb{R}[X]/(X)}$ ⁠,⁠${\displaystyle \mathbb{R}[X]/(X+1)}$ ⁠, and${\displaystyle \mathbb{R}[X]/(X-1)}$  are all isomorphic to${\displaystyle \mathbb{R}}$  and gain little interest at first. But note that${\displaystyle \mathbb{R}[X]/(X^2)}$  is called thedual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of${\displaystyle \mathbb{R}[X]}$  by⁠${\displaystyle X^2}$ ⁠. This variation of a complex plane arises as asubalgebra whenever the algebra contains areal line and anilpotent.

Furthermore, the ring quotient${\displaystyle \mathbb{R}[X]/(X^2-1)}$  does split into${\displaystyle \mathbb{R}[X]/(X+1)}$  and⁠${\displaystyle \mathbb{R}[X]/(X-1)}$ ⁠, so this ring is often viewed as thedirect sum⁠${\displaystyle \mathbb{R}\oplus \mathbb{R}}$ ⁠.Nevertheless, a variation on complex numbers${\displaystyle z=x+yj}$  is suggested by${\displaystyle j}$  as a root of⁠${\displaystyle X^2-1=0}$ ⁠, compared to${\displaystyle i}$  as root of⁠${\displaystyle X^2+1=0}$ ⁠. This plane ofsplit-complex numbers normalizes the direct sum${\displaystyle \mathbb{R}\oplus \mathbb{R}}$  by providing a basis${\displaystyle \left\{1,j\right\}}$  for 2-space where the identity of the algebra is at unit distance from the zero. With this basis aunit hyperbola may be compared to theunit circle of theordinary complex plane.

Suppose${\displaystyle X}$  and${\displaystyle Y}$  are two non-commutingindeterminates and form thefree algebra⁠${\displaystyle \mathbb{R}⟨X,Y⟩}$ ⁠. Then Hamilton'squaternions of 1843 can be cast as:$${\displaystyle \mathbb{R}⟨X,Y⟩/(X^2+1,Y^2+1,XY+YX)}$$ 

If${\displaystyle Y^2-1}$  is substituted for⁠${\displaystyle Y^2+1}$ ⁠, then one obtains the ring ofsplit-quaternions. Theanti-commutative property${\displaystyle YX=-XY}$  implies that${\displaystyle XY}$  has as its square:$${\displaystyle (XY)(XY)=X(YX)Y=-X(XY)Y=-(XX)(YY)=-(-1)(+1)=+1}$$ 

Substituting minus for plus inboth the quadratic binomials also results in split-quaternions.

The three types ofbiquaternions can also be written as quotients by use of the free algebra with three indeterminates${\displaystyle \mathbb{R}⟨X,Y,Z⟩}$  and constructing appropriate ideals.

Clearly, if${\displaystyle R}$  is acommutative ring, then so is⁠${\displaystyle R/I}$ ⁠; the converse, however, is not true in general.

The natural quotient map${\displaystyle p}$  has${\displaystyle I}$  as itskernel; 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${\displaystyle R/I}$  are essentially the same as the ring homomorphisms defined on${\displaystyle R}$  that vanish (i.e. are zero) on⁠${\displaystyle I}$ ⁠. More precisely, given a two-sided ideal${\displaystyle I}$  in${\displaystyle R}$  and a ring homomorphism${\displaystyle f:R\to S}$  whose kernel contains⁠${\displaystyle I}$ ⁠, there exists precisely one ring homomorphism${\displaystyle g:R/I\to S}$  with${\displaystyle gp=f}$  (where${\displaystyle p}$  is the natural quotient map). The map${\displaystyle g}$  here is given by the well-defined rule${\displaystyle g([a])=f(a)}$  for all${\displaystyle a}$  in⁠${\displaystyle 1R}$ ⁠. Indeed, thisuniversal property can be used todefine quotient rings and their natural quotient maps.

As a consequence of the above, one obtains the fundamental statement: every ring homomorphism${\displaystyle f:R\to S}$  induces aring isomorphism between the quotient ring${\displaystyle R/\ker (f)}$  and the image⁠${\displaystyle \mathrm{i}\mathrm{m}(f)}$ ⁠. (See also:Fundamental theorem on homomorphisms.)

The ideals of${\displaystyle R}$  and${\displaystyle R/I}$  are closely related: the natural quotient map provides abijection between the two-sided ideals of${\displaystyle R}$  that contain${\displaystyle I}$  and the two-sided ideals of${\displaystyle R/I}$  (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${\displaystyle M}$  is a two-sided ideal in${\displaystyle R}$  that contains⁠${\displaystyle I}$ ⁠, and we write${\displaystyle M/I}$  for the corresponding ideal in${\displaystyle R/I}$  (i.e.⁠${\displaystyle M/I=p(M)}$ ⁠), the quotient rings${\displaystyle R/M}$  and${\displaystyle (R/I)/(M/I)}$  are naturally isomorphic via the (well-defined) mapping⁠${\displaystyle a+M\mapsto (a+I)+M/I}$ ⁠.

The following facts prove useful incommutative algebra andalgebraic geometry: for${\displaystyle R\ne \{0\}}$  commutative,${\displaystyle R/I}$  is afield if and only if${\displaystyle I}$  is amaximal ideal, while${\displaystyle R/I}$  is anintegral domain if and only if${\displaystyle I}$  is aprime ideal. A number of similar statements relate properties of the ideal${\displaystyle I}$  to properties of the quotient ring⁠${\displaystyle R/I}$ ⁠.

TheChinese remainder theorem states that, if the ideal${\displaystyle I}$  is the intersection (or equivalently, the product) of pairwisecoprime ideals⁠${\displaystyle I_1,\dots ,I_k}$ ⁠, then the quotient ring${\displaystyle R/I}$  is isomorphic to theproduct of the quotient rings⁠${\displaystyle R/I_n,n=1,\dots ,k}$ ⁠.

Anassociative algebra${\displaystyle A}$  over acommutative ring${\displaystyle R}$  is a ring itself. If${\displaystyle I}$  is an ideal in${\displaystyle A}$  (closed under${\displaystyle R}$ -multiplication), then${\displaystyle A/I}$  inherits the structure of an algebra over${\displaystyle R}$  and is thequotient algebra.

• Associated graded ring
• Residue field
• Goldie's theorem
• Quotient module

• F. Kasch (1978)Moduln und Ringe, translated by DAR Wallace (1982)Modules and Rings,Academic Press, page 33.
• Neal H. McCoy (1948)Rings and Ideals, §13 Residue class rings, page 61, Carus Mathematical Monographs #8,Mathematical Association of America.
• Joseph Rotman (1998).Galois Theory (2nd ed.). Springer. pp. 21–23.ISBN 0-387-98541-7.
• B.L. van der Waerden (1970)Algebra, translated by Fred Blum and John R Schulenberger, Frederick Ungar Publishing, New York. See Chapter 3.5, "Ideals. Residue Class Rings", pp. 47–51.

• "Quotient ring",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Ideals and factor rings from John Beachy'sAbstract Algebra Online