Zero-product property explained

In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words,

ifab=0,thena=0orb=0.

This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties.^[1] All of the number systems studied in elementary mathematics - the integers

, the rational numbers

, the real numbers

\Reals

, and the complex numbers

\Complex

- satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.

Algebraic context

Suppose

is an algebraic structure. We might ask, does

have the zero-product property? In order for this question to have meaning,

must have both additive structure and multiplicative structure.^[2] Usually one assumes that

is a ring, though it could be something else, e.g. the set of nonnegative integers

\{0,1,2,\ldots\}

with ordinary addition and multiplication, which is only a (commutative) semiring.

Note that if

satisfies the zero-product property, and if

is a subset of

, then

also satisfies the zero product property: if

and

are elements of

such that

ab=0

, then either

a=0

b=0

because

and

can also be considered as elements of

Examples

A ring in which the zero-product property holds is called a domain. A commutative domain with a multiplicative identity element is called an integral domain. Any field is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain. Thus, the zero-product property holds for any subring of a skew field.
If

is a prime number, then the ring of integers modulo

has the zero-product property (in fact, it is a field).

The Gaussian integers are an integral domain because they are a subring of the complex numbers.
In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative.
The set of nonnegative integers

\{0,1,2,\ldots\}

is not a ring (being instead a semiring), but it does satisfy the zero-product property.

Non-examples

\Z_n

denote the ring of integers modulo

. Then

\Z₆

does not satisfy the zero product property: 2 and 3 are nonzero elements, yet

2 ⋅ 3\equiv0\pmod{6}

In general, if

is a composite number, then

\Z_n

does not satisfy the zero-product property. Namely, if

n=qm

where

0<q,m<n

, then

and

are nonzero modulo

, yet

qm\equiv0\pmod{n}

The ring

\Z²

of 2×2 matrices with integer entries does not satisfy the zero-product property: if

M = \begin1 & -1 \\ 0 & 0\end

and

N=\begin{pmatrix}0&1\ 0&1\end{pmatrix},

then

MN = \begin1 & -1 \\ 0 & 0\end \begin0 & 1 \\ 0 & 1\end = \begin0 & 0 \\ 0 & 0\end = 0,

yet neither

nor

is zero.

The ring of all functions

f:[0,1]\to\R

, from the unit interval to the real numbers, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any n ≥ 2, functions

f_1,\ldots,f_n

, none of which is identically zero, such that

f_if_j

is identically zero whenever

i ≠ j

The same is true even if we consider only continuous functions, or only even infinitely smooth functions. On the other hand, analytic functions have the zero-product property.

Application to finding roots of polynomials

Suppose

and

are univariate polynomials with real coefficients, and

is a real number such that

P(x)Q(x)=0

. (Actually, we may allow the coefficients and

to come from any integral domain.) By the zero-product property, it follows that either

P(x)=0

Q(x)=0

. In other words, the roots of

are precisely the roots of

together with the roots of

Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial

x³-2x²-5x+6

factorizes as

(x-3)(x-1)(x+2)

; hence, its roots are precisely 3, 1, and −2.

In general, suppose

is an integral domain and

is a monic univariate polynomial of degree

d\geq1

with coefficients in

. Suppose also that

has

distinct roots

r_1,\ldots,r_d\inR

. It follows (but we do not prove here) that

factorizes as

f(x)=(x-r₁₎ … (x-r_d)

. By the zero-product property, it follows that

r_1,\ldots,r_d

are the only roots of

: any root of

must be a root of

(x-r_i)

for some

. In particular,

has at most

distinct roots.

If however

is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial

x³+3x²+2x

has six roots in

\Z₆

(though it has only three roots in

References

David S. Dummit and Richard M. Foote, Abstract Algebra (3d ed.), Wiley, 2003, .

External links

PlanetMath: Zero rule of product

Notes and References

The other being a⋅0 = 0⋅a = 0. Mustafa A. Munem and David J. Foulis, Algebra and Trigonometry with Applications (New York: Worth Publishers, 1982), p. 4.
There must be a notion of zero (the additive identity) and a notion of products, i.e., multiplication.