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.[1] 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

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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   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

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  • 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

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  • 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

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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

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Notes

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  1. ^ 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.
  2. ^ There must be a notion of zero (the additive identity) and a notion of products, i.e., multiplication.

References

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