Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring<ref>Template:Cite book</ref> or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring <math>R</math> and a two-sided ideal <math>I</math> in Template:Tmath, a new ring, the quotient ring Template:Tmath, is constructed, whose elements are the cosets of <math>I</math> in <math>R</math> subject to special <math>+</math> and <math>\cdot</math> operations. (Quotient ring notation always uses a fraction slash "Template:Tmath".)

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 <math>R</math> and a two-sided ideal <math>I</math> in Template:Tmath, we may define an equivalence relation <math>\sim</math> on <math>R</math> as follows:

<math>a \sim b</math> if and only if <math>a - b</math> is in Template:Tmath.

Using the ideal properties, it is not difficult to check that <math>\sim</math> is a congruence relation. In case Template:Tmath, we say that <math>a</math> and <math>b</math> are congruent modulo <math>I</math> (for example, <math>1</math> and <math>3</math> are congruent modulo <math>2</math> as their difference is an element of the ideal Template:Tmath, the even integers). The equivalence class of the element <math>a</math> in <math>R</math> is given by: <math display="block">\left[ a \right] = a + I := \left\lbrace a + r : r \in I \right\rbrace</math>

This equivalence class is also sometimes written as <math>a \bmod I</math> and called the "residue class of <math>a</math> modulo <math>I</math>".

The set of all such equivalence classes is denoted by Template:Tmath; it becomes a ring, the factor ring or quotient ring of <math>R</math> modulo Template:Tmath, if one defines

• Template:Tmath;
• Template:Tmath.

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of <math>R\ /\ I</math> is Template:Tmath, and the multiplicative identity is Template:Tmath.

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

Examples

• The quotient ring <math>R\ /\ \lbrace 0 \rbrace</math> is naturally isomorphic to Template:Tmath, and <math>R / R</math> is the zero ring Template:Tmath, since, by our definition, for any Template:Tmath, we have that Template:Tmath, which equals <math>R</math> itself. This fits with the rule of thumb that the larger the ideal Template:Tmath, the smaller the quotient ring Template:Tmath. If <math>I</math> is a proper ideal of Template:Tmath, i.e., Template:Tmath, then <math>R / I</math> is not the zero ring.
• Consider the ring of integers <math>\mathbb{Z}</math> and the ideal of even numbers, denoted by Template:Tmath. Then the quotient ring <math>\mathbb{Z} / 2 \mathbb{Z}</math> has only two elements, the coset <math>0 + 2 \mathbb{Z}</math> consisting of the even numbers and the coset <math>1 + 2 \mathbb{Z}</math> consisting of the odd numbers; applying the definition, Template:Tmath, where <math>2 \mathbb{Z}</math> is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, Template:Tmath. Intuitively: if you think of all the even numbers as Template:Tmath, then every integer is either <math>0</math> (if it is even) or <math>1</math> (if it is odd and therefore differs from an even number by Template:Tmath). Modular arithmetic is essentially arithmetic in the quotient ring <math>\mathbb{Z} / n \mathbb{Z}</math> (which has <math>n</math> elements).
• Now consider the ring of polynomials in the variable <math>X</math> with real coefficients, Template:Tmath, and the ideal <math>I = \left( X^2 + 1 \right)</math> consisting of all multiples of the polynomial Template:Tmath. The quotient ring <math>\mathbb{R} [X]\ /\ ( X^2 + 1 )</math> is naturally isomorphic to the field of complex numbers Template:Tmath, with the class <math>[X]</math> playing the role of the imaginary unit Template:Tmath. The reason is that we "forced" Template:Tmath, i.e. Template:Tmath, which is the defining property of Template:Tmath. Since any integer exponent of <math>i</math> must be either <math>\pm i</math> or Template:Tmath, that means all possible polynomials essentially simplify to the form Template:Tmath. (To clarify, the quotient ring Template:Tmath is actually naturally isomorphic to the field of all linear polynomials Template:Tmath, where the operations are performed modulo Template:Tmath. In return, we have Template:Tmath, and this is matching <math>X</math> 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 <math>K</math> is some field and <math>f</math> is an irreducible polynomial in Template:Tmath. Then <math>L = K[X]\ /\ (f)</math> is a field whose minimal polynomial over <math>K</math> is Template:Tmath, which contains <math>K</math> as well as an element Template:Tmath.
• One important instance of the previous example is the construction of the finite fields. Consider for instance the field <math>F_3 = \mathbb{Z} / 3\mathbb{Z}</math> with three elements. The polynomial <math>f(X) = Xi^2 +1</math> is irreducible over <math>F_3</math> (since it has no root), and we can construct the quotient ring Template:Tmath. This is a field with <math>3^2 = 9</math> elements, denoted by Template:Tmath. 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 <math>V = \left\lbrace (x,y) | x^2 = y^3 \right\rbrace</math> as a subset of the real plane Template:Tmath. The ring of real-valued polynomial functions defined on <math>V</math> can be identified with the quotient ring Template:Tmath, and this is the coordinate ring of Template:Tmath. The variety <math>V</math> is now investigated by studying its coordinate ring.
• Suppose <math>M</math> is a <math>\mathbb{C}^{\infty}</math>-manifold, and <math>p</math> is a point of Template:Tmath. Consider the ring <math>R = \mathbb{C}^{\infty}(M)</math> of all <math>\mathbb{C}^{\infty}</math>-functions defined on <math>M</math> and let <math>I</math> be the ideal in <math>R</math> consisting of those functions <math>f</math> which are identically zero in some neighborhood <math>U</math> of <math>p</math> (where <math>U</math> may depend on Template:Tmath). Then the quotient ring <math>R\ /\ I</math> is the ring of germs of <math>\mathbb{C}^{\infty}</math>-functions on <math>M</math> at Template:Tmath.
• Consider the ring <math>F</math> of finite elements of a hyperreal field Template:Tmath. It consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers <math>x</math> for which a standard integer <math>n</math> with <math>-n < x < n</math> exists. The set <math>I</math> of all infinitesimal numbers in Template:Tmath, together with Template:Tmath, is an ideal in Template:Tmath, and the quotient ring <math>F\ /\ I</math> is isomorphic to the real numbers Template:Tmath. The isomorphism is induced by associating to every element <math>x</math> of <math>F</math> the standard part of Template:Tmath, i.e. the unique real number that differs from <math>x</math> by an infinitesimal. In fact, one obtains the same result, namely Template:Tmath, if one starts with the ring <math>F</math> of finite hyperrationals (i.e. ratio of a pair of hyperintegers), see construction of the real numbers.

Variations of complex planes

The quotients Template:Tmath, Template:Tmath, and <math>\mathbb{R} [X] / (X - 1)</math> are all isomorphic to <math>\mathbb{R}</math> and gain little interest at first. But note that <math>\mathbb{R} [X] / (X^2)</math> is called the dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of <math>\mathbb{R} [X]</math> by Template:Tmath. This variation of a complex plane arises as a subalgebra whenever the algebra contains a real line and a nilpotent.

Furthermore, the ring quotient <math>\mathbb{R} [X] / (X^2 - 1)</math> does split into <math>\mathbb{R} [X] / (X + 1)</math> and Template:Tmath, so this ring is often viewed as the direct sum Template:Tmath. Nevertheless, a variation on complex numbers <math>z = x + yj</math> is suggested by <math>j</math> as a root of Template:Tmath, compared to <math>i</math> as root of Template:Tmath. This plane of split-complex numbers normalizes the direct sum <math>\mathbb{R} \oplus \mathbb{R}</math> by providing a basis <math>\left\lbrace 1, j \right\rbrace</math> 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 <math>X</math> and <math>Y</math> are two non-commuting indeterminates and form the free algebra Template:Tmath. Then Hamilton's quaternions of 1843 can be cast as: <math display="block">\mathbb{R} \langle X, Y \rangle / ( X^2 + 1,\, Y^2 + 1,\, XY + YX )</math>

If <math>Y^2 - 1</math> is substituted for Template:Tmath, then one obtains the ring of split-quaternions. The anti-commutative property <math>YX = -XY</math> implies that <math>XY</math> has as its square: <math display="block">(XY) (XY) = X (YX) Y = -X (XY) Y = -(XX) (YY) = -(-1)(+1) = +1</math>

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 <math>\mathbb{R} \langle X, Y, Z \rangle</math> and constructing appropriate ideals.

Properties

Clearly, if <math>R</math> is a commutative ring, then so is Template:Tmath; the converse, however, is not true in general.

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

The ideals of <math>R</math> and <math>R\ /\ I</math> are closely related: the natural quotient map provides a bijection between the two-sided ideals of <math>R</math> that contain <math>I</math> and the two-sided ideals of <math>R\ /\ I</math> (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 <math>M</math> is a two-sided ideal in <math>R</math> that contains Template:Tmath, and we write <math>M\ /\ I</math> for the corresponding ideal in <math>R\ /\ I</math> (i.e. Template:Tmath), the quotient rings <math>R\ /\ M</math> and <math>(R / I)\ /\ (M / I)</math> are naturally isomorphic via the (well-defined) mapping Template:Tmath.

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

The Chinese remainder theorem states that, if the ideal <math>I</math> is the intersection (or equivalently, the product) of pairwise coprime ideals Template:Tmath, then the quotient ring <math>R\ /\ I</math> is isomorphic to the product of the quotient rings Template:Tmath.

For algebras over a ring

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

See also

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

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.
• Template:Cite book
• 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

• Template:Springer
• Ideals and factor rings from John Beachy's Abstract Algebra Online