Zero-product property

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In the mathematical areas of algebra and analysis, the zero-product property is an abstract and explicit statement of the familiar property from everyday mathematics that if the product of two real numbers is zero, then at least one of the numbers in the product (factors) must be zero.

1 Informal statement of the zero-product property
2 Introduction
3 Algebraic context
4 The zero-product property
5 Examples
6 Examples of structures which do not possess the zero-product property
7 Zero divisors
8 Uses in analysis
9 Proofs
- 9.1 a × 0 = 0
- 9.2 Z(R) is prime ideal of R
10 See also
11 External links
12 References

[edit] Informal statement of the zero-product property

For all numbers a and b, ab = 0 implies a = 0 or b = 0 (or both).

[edit] Introduction

Elementary mathematics includes the study of the sets of natural numbers, the integers, the rational numbers, and the real numbers. In each of these there is a number (or element) 0 which has two properties:

that 0 multiplied by any number a is 0: that is, 0 × a = 0 = a × 0 for any a in the set;
if the product of two numbers a and b is 0 then one or both of a or b must be zero; otherwise stated, the product of two non-zero numbers is non-zero. Either of these is the zero-product property.

From Property 1 we derive the fact that, for example, 2 × 0 = 0. From Property 2 we know that if 2x = 0, where x is a real number, then we can be assured that x = 0. This is a result of the fact that the zero-product property holds for the real numbers.

[edit] Algebraic context

The study of algebra involves consideration of sets of elements and operations on them. There are often elements in a set which possess special properties. For example, if A is a group with operation + then there is a unique element 0 in A, called the additive identity, such that

0 + a = a = a + 0 for all a in A.

If the operation × of multiplication is defined for A (such as in the mathematical object known as a ring) then we can define an element 1 in A, called the multiplicative identity or unity, such that

1 × a = a = a × 1 for all a in A.

The interplay between the additive and multiplicative operations (where multiplication distributes over addition) leads automatically to Property 1: that 0 × a = 0 = a × 0 for all a in A (which is proved below). This is true for any context in which addition and multiplication have group structures defined on the same set (for example, an algebra). Property 2 is, however, not a natural consequence of this interplay, as there are algebraic structures in which addition and multiplication are defined which do not have the zero-product property.

[edit] The zero-product property

The formal statement of the zero-product property is:

Let A be a ring. The zero-product property holds in A if for all elements a and b in A we have ab = 0 implies a = 0 or b = 0.

[edit] Examples

An integral domain is a ring in which, by definition, the zero-product property holds. Thus, for example, the zero-product property holds in the ring of integers modulo p,

Z_p = {0, 1, 2, ..., p − 1}

where p is a prime, the integers Z, the rational numbers Q, the real numbers R, and the complex numbers C since these are all integral domains.

Since any field is an integral domain, the zero-product property holds in any field.

[edit] Examples of structures which do not possess the zero-product property

It is not, however, true that the zero-product property necessarily holds in every structure which possesses an additive and multiplicative operation.

If R is not an integral domain, then the zero-product property does not hold: that is, there are non-zero a and b in R such that ab = 0. For a specific example of this, if R is the ring of integers modulo 6, or Z₆ = {0, 1, 2, 3, 4, 5} then there are 2, 3 in Z₆ such that 2 × 3 ≡ 0 (mod) 6 but neither 2 ≡ 0 (mod 6) nor 3 ≡ 0 (mod 6).
In general, Z_n, where n is a composite number, is not an integral domain and therefore the zero-product property does not hold. For if

n = m × q

with m, q < n, then neither m nor q is equal to 0 (mod n), but m × q = n = 0 (mod n), violating the zero-product property.

If R is the ring M₂(Z) of 2 by 2 matrices with integer coefficients then there are matrices

$M = \begin{pmatrix}1 & -1 \\ 0 & 0\end{pmatrix}$ and $N = \begin{pmatrix}0 & 1 \\ 0 & 1\end{pmatrix}$

neither of which is equal to the zero matrix

$0 = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}$

such that

$MN = \begin{pmatrix}1 & -1 \\ 0 & 0\end{pmatrix} \begin{pmatrix}0 & 1 \\ 0 & 1\end{pmatrix} = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix} = 0$ ,

so M₂(Z) does not possess the zero-product property.

[edit] Zero divisors

In a ring R the set of elements for which the zero-product rule does not hold are called the zero divisors of the ring, that is the set

Z(R) = {r in R : there is an s in R such that rs = 0}.

Thus the zero divisors of a ring R are a measure of how much the zero-product property holds in R.

0 is always a zero divisor.

The zero divisors of an integral domain consist only of the element 0, or Z(R) = {0}.

Since the zero divisors Z(R) of a ring R form a prime ideal of R (proved below), it follows that the quotient of R by Z(R), or R/Z(R) is an integral domain. That is to say, by forming equivalence classes based on all elements on which the zero-product property does not hold, one obtains a ring structure in which the zero-product property holds for all elements in the quotient.

[edit] Uses in analysis

The zero-product property is used in solving polynomial equations over the real numbers, such as quadratic equations.

For example, when finding all values of x which satisfy x² + x − 6 = 0, we first factor the left side of the equation to obtain (x + 3)(x − 2) = 0. Then, by the zero-product property, we know that either x + 3 = 0 (in which case x = − 3), or x − 2 = 0 (in which case x = 2). Thus, our solution is all x in the set {−3, 2}.

This method can be extended to polynomials of higher degree. In general, if a polynomial of degree n can be written as a product of factors (x − a₁)(x − a₂) ... (x − a_n) = 0 then the solution set is {a₁, a₂, ..., a_n}.

In the complex numbers the extension given above applies to any polynomial. Thus a polynomial f(z) of degree n over C can be written as a product of n factors (z − α) where α is one of precisely n roots of f. This is known as the fundamental theorem of algebra.

Note that solving quadratic equations in algebraic structures in which the zero-product property does not hold can lead to surprising results. For example, the quadratic equation

x² − x = 0

has solutions {0, 1} in Z, Q, or R, but in Z₆ the solution set is {0, 1, 3, 4} since 3² − 3 = 6 ≡ 0 (mod 6) and 4² − 4 = 12 ≡ 0 (mod 6).

[edit] Proofs

[edit] a × 0 = 0

Let A be ring with more than one element and let a be a non-zero element of A. Then a × 0 = a × (a + −a) = a² + (−a²) = 0.

[edit] Z(R) is prime ideal of R

Let R be a commutative ring. Sums and products of zero divisors are themselves zero divisors, so Z(R) is a subring of R. If a is a non-zero element of Z(R) then there is a non-zero c in R such that ac = 0. If b is any element of R then

0 = (ac)b = (ab)c

whence ab is an element of Z(R), making Z(R) an ideal of R. Let a and b be elements of R and suppose that ab is an element of Z(R). It follows that there is a c in R such that (ab)c = 0. By associativity we can write a(bc) = 0 and thus a is an element of Z(R), making Z(R) a prime ideal of R.

[edit] See also

[edit] External links

PlanetMath: Zero rule of product

[edit] References

David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3d ed.): 2003, ISBN 0-471-43334-9.

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