Zero-product property

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

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.  All of the number systems studied in elementary mathematics — the integers , the rational numbers , the real numbers , and the complex numbers  — 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.  Usually one assumes that  is a ring, though it could be something else, e.g. the set of nonnegative integers  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 , then either  or  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  is not a ring (being instead a semiring), but it does satisfy the zero-product property.

Non-examples

 Let  denote the ring of integers modulo . Then  does not satisfy the zero product property: 2 and 3 are nonzero elements, yet .
 In general, if  is a composite number, then  does not satisfy the zero-product property.  Namely, if  where , then  and  are nonzero modulo , yet .
 The ring  of 2×2 matrices with integer entries does not satisfy the zero-product property: if  and  then  yet neither  nor  is zero.
 The ring of all functions , 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 , none of which is identically zero, such that  is identically zero whenever .
 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 .  (Actually, we may allow the coefficients and  to come from any integral domain.)  By the zero-product property, it follows that either  or .  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  factorizes as ; hence, its roots are precisely 3, 1, and −2.

In general, suppose  is an integral domain and  is a monic univariate polynomial of degree  with coefficients in .  Suppose also that  has  distinct roots .  It follows (but we do not prove here) that  factorizes as .  By the zero-product property, it follows that  are the only roots of : any root of  must be a root of  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  has six roots in  (though it has only three roots in ).

See also
 Fundamental theorem of algebra
 Integral domain and domain
 Prime ideal
 Zero divisor

Notes

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

External links
 PlanetMath: Zero rule of product

Abstract algebra
Elementary algebra
Real analysis
Ring theory
0 (number)