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Formal quotient ring construction

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

Using the ideal properties, it is not difficult to check that ∼ {\displaystyle \sim } is a congruence relation. In case ⁠ a ∼ b {\displaystyle a\sim b} ⁠, we say that a {\displaystyle a} and b {\displaystyle b} are congruent modulo I {\displaystyle I} (for example, 1 {\displaystyle 1} and 3 {\displaystyle 3} are congruent modulo 2 {\displaystyle 2} as their difference is an element of the ideal ⁠ 2 Z {\displaystyle 2\mathbb {Z} } ⁠, the even integers). The equivalence class of the element a {\displaystyle a} in R {\displaystyle R} is given by: [ a ] = a + I := { a + r : r ∈ I } {\displaystyle \left[a\right]=a+I:=\left\lbrace a+r:r\in I\right\rbrace }

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

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

• ⁠ ( a + I ) + ( b + I ) = ( a + b ) + I {\displaystyle (a+I)+(b+I)=(a+b)+I} ⁠;
• ⁠ ( 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 R   /   I {\displaystyle R\ /\ I} is ⁠ 0 ¯ = ( 0 + I ) = I {\displaystyle {\bar {0}}=(0+I)=I} ⁠, and the multiplicative identity is ⁠ 1 ¯ = ( 1 + I ) {\displaystyle {\bar {1}}=(1+I)} ⁠.

The map p {\displaystyle p} from R {\displaystyle R} to R   /   I {\displaystyle R\ /\ I} defined by p ( a ) = a + I {\displaystyle 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 } {\displaystyle R\ /\ \lbrace 0\rbrace } is naturally isomorphic to ⁠ R {\displaystyle R} ⁠, and R / R {\displaystyle R/R} is the zero ring ⁠ { 0 } {\displaystyle \lbrace 0\rbrace } ⁠, since, by our definition, for any ⁠ r ∈ R {\displaystyle r\in R} ⁠, we have that ⁠ [ r ] = r + R = { r + b : b ∈ R } {\displaystyle \left[r\right]=r+R=\left\lbrace r+b:b\in R\right\rbrace } ⁠, which equals R {\displaystyle R} itself. This fits with the rule of thumb that the larger the ideal ⁠ I {\displaystyle I} ⁠, the smaller the quotient ring ⁠ R   /   I {\displaystyle R\ /\ I} ⁠. If I {\displaystyle I} is a proper ideal of ⁠ R {\displaystyle R} ⁠, i.e., ⁠ I ≠ R {\displaystyle I\neq R} ⁠, then R / I {\displaystyle R/I} is not the zero ring.
• Consider the ring of integers Z {\displaystyle \mathbb {Z} } and the ideal of even numbers, denoted by ⁠ 2 Z {\displaystyle 2\mathbb {Z} } ⁠. Then the quotient ring Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } has only two elements, the coset 0 + 2 Z {\displaystyle 0+2\mathbb {Z} } consisting of the even numbers and the coset 1 + 2 Z {\displaystyle 1+2\mathbb {Z} } consisting of the odd numbers; applying the definition, ⁠ [ z ] = z + 2 Z = { z + 2 y : 2 y ∈ 2 Z } {\displaystyle \left[z\right]=z+2\mathbb {Z} =\left\lbrace z+2y:2y\in 2\mathbb {Z} \right\rbrace } ⁠, where 2 Z {\displaystyle 2\mathbb {Z} } is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, ⁠ F 2 {\displaystyle F_{2}} ⁠. Intuitively: if you think of all the even numbers as ⁠ 0 {\displaystyle 0} ⁠, then every integer is either 0 {\displaystyle 0} (if it is even) or 1 {\displaystyle 1} (if it is odd and therefore differs from an even number by ⁠ 1 {\displaystyle 1} ⁠). Modular arithmetic is essentially arithmetic in the quotient ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } (which has n {\displaystyle n} elements).
• Now consider the ring of polynomials in the variable X {\displaystyle X} with real coefficients, ⁠ R [ X ] {\displaystyle \mathbb {R} [X]} ⁠, and the ideal I = ( X 2 + 1 ) {\displaystyle I=\left(X^{2}+1\right)} consisting of all multiples of the polynomial ⁠ X 2 + 1 {\displaystyle X^{2}+1} ⁠. The quotient ring R [ X ]   /   ( X 2 + 1 ) {\displaystyle \mathbb {R} [X]\ /\ (X^{2}+1)} is naturally isomorphic to the field of complex numbers ⁠ C {\displaystyle \mathbb {C} } ⁠, with the class [ X ] {\displaystyle [X]} playing the role of the imaginary unit ⁠ i {\displaystyle i} ⁠. The reason is that we "forced" ⁠ X 2 + 1 = 0 {\displaystyle X^{2}+1=0} ⁠, i.e. ⁠ X 2 = − 1 {\displaystyle X^{2}=-1} ⁠, which is the defining property of ⁠ i {\displaystyle i} ⁠. Since any integer exponent of i {\displaystyle i} must be either ± i {\displaystyle \pm i} or ⁠ ± 1 {\displaystyle \pm 1} ⁠, that means all possible polynomials essentially simplify to the form ⁠ a + b i {\displaystyle a+bi} ⁠. (To clarify, the quotient ring ⁠ R [ X ]   /   ( X 2 + 1 ) {\displaystyle \mathbb {R} [X]\ /\ (X^{2}+1)} ⁠ is actually naturally isomorphic to the field of all linear polynomials ⁠ a X + b ; a , b ∈ R {\displaystyle aX+b;a,b\in \mathbb {R} } ⁠, where the operations are performed modulo ⁠ X 2 + 1 {\displaystyle X^{2}+1} ⁠. In return, we have ⁠ X 2 = − 1 {\displaystyle X^{2}=-1} ⁠, and this is matching X {\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 {\displaystyle K} is some field and f {\displaystyle f} is an irreducible polynomial in ⁠ K [ X ] {\displaystyle K[X]} ⁠. Then L = K [ X ]   /   ( f ) {\displaystyle L=K[X]\ /\ (f)} is a field whose minimal polynomial over K {\displaystyle K} is ⁠ f {\displaystyle f} ⁠, which contains K {\displaystyle K} as well as an element ⁠ x = X + ( f ) {\displaystyle 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 {\displaystyle F_{3}=\mathbb {Z} /3\mathbb {Z} } with three elements. The polynomial f ( X ) = X i 2 + 1 {\displaystyle f(X)=Xi^{2}+1} is irreducible over F 3 {\displaystyle F_{3}} (since it has no root), and we can construct the quotient ring ⁠ F 3 [ X ]   /   ( f ) {\displaystyle F_{3}[X]\ /\ (f)} ⁠. This is a field with 3 2 = 9 {\displaystyle 3^{2}=9} elements, denoted by ⁠ F 9 {\displaystyle 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 } {\displaystyle V=\left\lbrace (x,y)|x^{2}=y^{3}\right\rbrace } as a subset of the real plane ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠. The ring of real-valued polynomial functions defined on V {\displaystyle V} can be identified with the quotient ring ⁠ R [ X , Y ]   /   ( X 2 − Y 3 ) {\displaystyle \mathbb {R} [X,Y]\ /\ (X^{2}-Y^{3})} ⁠, and this is the coordinate ring of ⁠ V {\displaystyle V} ⁠. The variety V {\displaystyle V} is now investigated by studying its coordinate ring.
• Suppose M {\displaystyle M} is a C ∞ {\displaystyle \mathbb {C} ^{\infty }} -manifold, and p {\displaystyle p} is a point of ⁠ M {\displaystyle M} ⁠. Consider the ring R = C ∞ ( M ) {\displaystyle R=\mathbb {C} ^{\infty }(M)} of all C ∞ {\displaystyle \mathbb {C} ^{\infty }} -functions defined on M {\displaystyle M} and let I {\displaystyle I} be the ideal in R {\displaystyle R} consisting of those functions f {\displaystyle f} which are identically zero in some neighborhood U {\displaystyle U} of p {\displaystyle p} (where U {\displaystyle U} may depend on ⁠ f {\displaystyle f} ⁠). Then the quotient ring R   /   I {\displaystyle R\ /\ I} is the ring of germs of C ∞ {\displaystyle \mathbb {C} ^{\infty }} -functions on M {\displaystyle M} at ⁠ p {\displaystyle p} ⁠.
• Consider the ring F {\displaystyle F} of finite elements of a hyperreal field ⁠ ∗ R {\displaystyle ^{*}\mathbb {R} } ⁠. It consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers x {\displaystyle x} for which a standard integer n {\displaystyle n} with − n < x < n {\displaystyle -n<x<n} exists. The set I {\displaystyle I} of all infinitesimal numbers in ⁠ ∗ R {\displaystyle ^{*}\mathbb {R} } ⁠, together with ⁠ 0 {\displaystyle 0} ⁠, is an ideal in ⁠ F {\displaystyle F} ⁠, and the quotient ring F   /   I {\displaystyle F\ /\ I} is isomorphic to the real numbers ⁠ R {\displaystyle \mathbb {R} } ⁠. The isomorphism is induced by associating to every element x {\displaystyle x} of F {\displaystyle F} the standard part of ⁠ x {\displaystyle x} ⁠, i.e. the unique real number that differs from x {\displaystyle x} by an infinitesimal. In fact, one obtains the same result, namely ⁠ R {\displaystyle \mathbb {R} } ⁠, if one starts with the ring F {\displaystyle 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 ) {\displaystyle \mathbb {R} [X]/(X)} ⁠, ⁠ R [ X ] / ( X + 1 ) {\displaystyle \mathbb {R} [X]/(X+1)} ⁠, and R [ X ] / ( X − 1 ) {\displaystyle \mathbb {R} [X]/(X-1)} are all isomorphic to R {\displaystyle \mathbb {R} } and gain little interest at first. But note that R [ X ] / ( X 2 ) {\displaystyle \mathbb {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 ] {\displaystyle \mathbb {R} [X]} by ⁠ X 2 {\displaystyle 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 ) {\displaystyle \mathbb {R} [X]/(X^{2}-1)} does split into R [ X ] / ( X + 1 ) {\displaystyle \mathbb {R} [X]/(X+1)} and ⁠ R [ X ] / ( X − 1 ) {\displaystyle \mathbb {R} [X]/(X-1)} ⁠, so this ring is often viewed as the direct sum ⁠ R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } ⁠. Nevertheless, a variation on complex numbers z = x + y j {\displaystyle z=x+yj} is suggested by j {\displaystyle j} as a root of ⁠ X 2 − 1 = 0 {\displaystyle X^{2}-1=0} ⁠, compared to i {\displaystyle i} as root of ⁠ X 2 + 1 = 0 {\displaystyle X^{2}+1=0} ⁠. This plane of split-complex numbers normalizes the direct sum R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } by providing a basis { 1 , j } {\displaystyle \left\lbrace 1,j\right\rbrace } 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 {\displaystyle X} and Y {\displaystyle Y} are two non-commuting indeterminates and form the free algebra ⁠ R ⟨ X , Y ⟩ {\displaystyle \mathbb {R} \langle X,Y\rangle } ⁠. Then Hamilton's quaternions of 1843 can be cast as: R ⟨ X , Y ⟩ / ( X 2 + 1 , Y 2 + 1 , X Y + Y X ) {\displaystyle \mathbb {R} \langle X,Y\rangle /(X^{2}+1,\,Y^{2}+1,\,XY+YX)}

If Y 2 − 1 {\displaystyle Y^{2}-1} is substituted for ⁠ Y 2 + 1 {\displaystyle Y^{2}+1} ⁠, then one obtains the ring of split-quaternions. The anti-commutative property Y X = − X Y {\displaystyle YX=-XY} implies that X Y {\displaystyle XY} has as its square: ( X Y ) ( X Y ) = X ( Y X ) Y = − X ( X Y ) Y = − ( X X ) ( Y Y ) = − ( − 1 ) ( + 1 ) = + 1 {\displaystyle (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 ⟩ {\displaystyle \mathbb {R} \langle X,Y,Z\rangle } and constructing appropriate ideals.

Properties

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

The natural quotient map p {\displaystyle p} has I {\displaystyle 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 {\displaystyle R\ /\ I} are essentially the same as the ring homomorphisms defined on R {\displaystyle R} that vanish (i.e. are zero) on ⁠ I {\displaystyle I} ⁠. More precisely, given a two-sided ideal I {\displaystyle I} in R {\displaystyle R} and a ring homomorphism f : R → S {\displaystyle f:R\to S} whose kernel contains ⁠ I {\displaystyle I} ⁠, there exists precisely one ring homomorphism g : R   /   I → S {\displaystyle g:R\ /\ I\to S} with g p = f {\displaystyle gp=f} (where p {\displaystyle p} is the natural quotient map). The map g {\displaystyle g} here is given by the well-defined rule g ( [ a ] ) = f ( a ) {\displaystyle g([a])=f(a)} for all a {\displaystyle a} in ⁠ 1 R {\displaystyle 1R} ⁠. 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 {\displaystyle f:R\to S} induces a ring isomorphism between the quotient ring R   /   ker ⁡ ( f ) {\displaystyle R\ /\ \ker(f)} and the image ⁠ i m ( f ) {\displaystyle \mathrm {im} (f)} ⁠. (See also: Fundamental theorem on homomorphisms.)

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

The following facts prove useful in commutative algebra and algebraic geometry: for R ≠ { 0 } {\displaystyle R\neq \lbrace 0\rbrace } commutative, R   /   I {\displaystyle R\ /\ I} is a field if and only if I {\displaystyle I} is a maximal ideal, while R / I {\displaystyle R/I} is an integral domain if and only if I {\displaystyle I} is a prime ideal. A number of similar statements relate properties of the ideal I {\displaystyle I} to properties of the quotient ring ⁠ R   /   I {\displaystyle R\ /\ I} ⁠.

The Chinese remainder theorem states that, if the ideal I {\displaystyle I} is the intersection (or equivalently, the product) of pairwise coprime ideals ⁠ I 1 , … , I k {\displaystyle I_{1},\ldots ,I_{k}} ⁠, then the quotient ring R   /   I {\displaystyle R\ /\ I} is isomorphic to the product of the quotient rings ⁠ R   /   I n , n = 1 , … , k {\displaystyle R\ /\ I_{n},\;n=1,\ldots ,k} ⁠.

For algebras over a ring

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

See also

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

Notes

Further references

• 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.

External links

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

References

1. Jacobson, Nathan (1984). Structure of Rings (revised ed.). American Mathematical Soc. ISBN 0-821-87470-5. 0-821-87470-5 ↩

2. Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. 0-471-43334-9 ↩

3. Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. 0-387-95385-X ↩