Zero-product property

Last updated

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

Contents

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

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

Non-examples

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

Notes

  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.

Related Research Articles

In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

<span class="mw-page-title-main">Prime ideal</span> Ideal in a ring which has properties similar to prime elements

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 the zero ideal.

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

<i>p</i>-adic number Number system extending the rational numbers

In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right.

In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, or equivalently if the map from R to R that sends x to ax is not injective. Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element a that is both a left and a right zero divisor is called a two-sided zero divisor. If the ring is commutative, then the left and right zero divisors are the same.

In mathematics, a unique factorization domain (UFD) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field of rational numbers from the ring of integers.

In algebra, a monic polynomial is a non-zero univariate polynomial in which the leading coefficient is equal to 1. That is to say, a monic polynomial is one that can be written as

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of .

In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. Equivalently, a domain is a ring in which 0 is the only left zero divisor. A commutative domain is called an integral domain. Mathematical literature contains multiple variants of the definition of "domain".

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

In abstract algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x−1 belongs to D.

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest positive number of copies of the ring's multiplicative identity (1) that will sum to the additive identity (0). If no such number exists, the ring is said to have characteristic zero.

In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.

In mathematics, an algebraic number field is an extension field of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .

References