If , then or .
The zero-product property is also known as the rule of zero product, the null factor law or the nonexistence of nontrivial zero divisors. 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.
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.[note 1] Usually one assumes that is a ring, though it could be something else, e.g., the nonnegative integers .
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 .
- 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, but it does satisfy the zero-product property.
- 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 by 2 matrices with integer entries does not satisfy the zero-product property: if
- and ,
- 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.
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 ).
- There must be a notion of zero (the additive identity) and a notion of products, i.e., multiplication.
- David S. Dummit and Richard M. Foote, Abstract Algebra (3d ed.), Wiley, 2003, ISBN 0-471-43334-9.