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, it is the following assertion:

If ab = 0, then a=0 or b=0.

The zero-product property is also known as the rule of zero product or nonexistence of nontrivial zero divisors. All of the number systems studied in elementary mathematics — the integers \mathbb{Z}, the rational numbers \mathbb{Q}, the real numbers \mathbb{R}, and the complex numbers \mathbb{C} — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.

Algebraic context[edit]

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.[note 1] Usually one assumes that A is a ring, though it could be something else, e.g., the nonnegative integers \{0,1,2,\ldots\}.

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[edit]

  • 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.
  • 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, but it does satisfy the zero-product property.

Non-examples[edit]

  • Let \mathbb{Z}_n denote the ring of integers modulo n. Then \mathbb{Z}_6 does not satisfy the zero product property: 2 and 3 are nonzero elements, yet 2 \cdot 3 \equiv 0 \pmod{6}.
  • In general, if n is a composite number, then \mathbb{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{n}.
  • The ring \mathbb{Z}^{2 \times 2} of 2 by 2 matrices with integer entries does not satisfy the zero-product property: if
M = \begin{pmatrix}1 & -1 \\ 0 & 0\end{pmatrix} and N = \begin{pmatrix}0 & 1 \\ 0 & 1\end{pmatrix},
then
MN = \begin{pmatrix}1 & -1 \\ 0 & 0\end{pmatrix} \begin{pmatrix}0 & 1 \\ 0 & 1\end{pmatrix} = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix} = 0,
yet neither M nor N is zero.
  • The ring of all functions f: [0,1] \to \mathbb{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 functions.

Application to finding roots of polynomials[edit]

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 \mathbb{Z}_6 (though it has only three roots in \mathbb{Z}).

See also[edit]

Notes[edit]

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

References[edit]

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

External links[edit]