Zero-product Property
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In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, \textab=0,\texta=0\textb=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. All of the number systems studied in elementary mathematics — the integers \Z, the rational numbers \Q, 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 A is an algebraic structure. We might ask, does A have the zero-product property? In order for this question to have meaning, A must have both additive structure and multiplicative structure.There must be a notion of zero (the additive identity) and a notion of products, i.e., multiplication. Usually one assumes that A 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 A satisfies the zero-product property, and if B is a subset of A, then B also satisfies the zero product property: if a and b are elements of B such that ab = 0, then either a = 0 or b = 0 because a and b can also be considered as elements of A.


Examples

* A ring in which the zero-product property holds is called a domain. A commutative domain with a
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
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 p is a prime number, then the ring of integers modulo p 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 \Z_n denote the ring of integers modulo n. Then \Z_6 does not satisfy the zero product property: 2 and 3 are nonzero elements, yet 2 \cdot 3 \equiv 0 \pmod. * In general, if n is a composite number, then \Z_n does not satisfy the zero-product property. Namely, if n = qm where 0 < q,m < n, then m and q are nonzero modulo n, yet qm \equiv 0 \pmod. * The ring \Z^ of 2×2 matrices with integer entries does not satisfy the zero-product property: if M = \begin1 & -1 \\ 0 & 0\end and N = \begin0 & 1 \\ 0 & 1\end, then MN = \begin1 & -1 \\ 0 & 0\end \begin0 & 1 \\ 0 & 1\end = \begin0 & 0 \\ 0 & 0\end = 0, yet neither M nor N is zero. * The ring of all functions f: ,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_i \, f_j is identically zero whenever i \neq j. * The same is true even if we consider only continuous functions, or only even infinitely
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s. On the other hand, analytic functions have the zero-product property.


Application to finding roots of polynomials

Suppose P and Q are univariate polynomials with real coefficients, and x is a real number such that P(x)Q(x) = 0. (Actually, we may allow the coefficients and x to come from any integral domain.) By the zero-product property, it follows that either P(x) = 0 or Q(x) = 0. In other words, the roots of PQ are precisely the roots of P together with the roots of Q. Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial x^3 - 2x^2 - 5x + 6 factorizes as (x-3)(x-1)(x+2); hence, its roots are precisely 3, 1, and −2. In general, suppose R is an integral domain and f is a monic univariate polynomial of degree d \geq 1 with coefficients in R. Suppose also that f has d distinct roots r_1,\ldots,r_d \in R. It follows (but we do not prove here) that f factorizes as f(x) = (x-r_1) \cdots (x-r_d). By the zero-product property, it follows that r_1,\ldots,r_d are the ''only'' roots of f: any root of f must be a root of (x-r_i) for some i. In particular, f has at most d distinct roots. If however R is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial x^3 + 3x^2 + 2x has six roots in \Z_6 (though it has only three roots in \Z).


See also

* Fundamental theorem of algebra * Integral domain and domain *
Prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
* Zero divisor


Notes


References

*David S. Dummit and Richard M. Foote, ''Abstract Algebra'' (3d ed.), Wiley, 2003, {{isbn, 0-471-43334-9.


External links


PlanetMath: Zero rule of product
Abstract algebra Elementary algebra Real analysis Ring theory 0 (number)