- Zero-product property
In the mathematical areas of algebra and analysis, the zero-product property, also known as the "zero-product rule", is an abstract and explicit statement of the familiar property from
elementary mathematics that if the product of tworeal number s is zero, then at least one of the numbers in the product (factors) must be zero.The zero-product property is the defining characteristic of a type of commutative ring called an
integral domain , in which every non-zero element is considered to possess this property.Informal statement of the zero-product property
* For all
number s "a" and "b", "ab" = 0 implies "a" = 0 or "b" = 0 (or both).Introduction
Elementary mathematics includes the study of the sets ofnatural number s, theinteger s, therational number s, and thereal number s. In each of these there is anumber (or element) zero which has two properties:# that 0 multiplied by any number "a" is 0: that is, 0 × "a" = 0 = "a" × 0 for any "a" in the set;
# if the product of two numbers "a" and "b" is 0 then one or both of "a" or "b" must be zero; otherwise stated, the product of two non-zero numbers is non-zero. Either of these is the zero-product property.From Property 1 we derive the fact that, for example, 2 × 0 = 0. From Property 2 we know that if 2"x" = 0, where "x" is a real number, then we can be assured that "x" = 0. This is a result of the fact that the zero-product property holds for the real numbers (and also for all of its subsets).
Elementary examples
The zero-product property is used in solving elementary
equation s in one variable.* In the solution of a
linear equation "ax" + "b" = 0 in thereal number s, where "a" is non-zero, we factorize by "a" to give "a"("x" + "b"/"a") = 0. By the zero-product property, since "a" is not zero, we conclude that "x" + "b"/"a" = 0, which implies that "x" = −"b"/"a".* In higher-degree
polynomial equation s the zero-product property may be invoked if the equation can by fully factored. See "Uses in analysis" below.Algebraic context
The study of algebra involves consideration of sets of elements and operations on them. There are often elements in a set which possess special properties. For example, if "A" is a group with operation + then there is a unique element 0 in "A", called the
additive identity , such that:0 + "a" = "a" = "a" + 0 for all "a" in "A".
If the operation × of
multiplication is defined for "A" (such as in the mathematical object known as a ring) then we can define an element 1 in "A", called themultiplicative identity or unity, such that:1 × "a" = "a" = "a" × 1 for all "a" in "A".
The interplay between the additive and multiplicative operations (where multiplication distributes over addition) leads automatically to Property 1: that 0 × "a" = 0 = "a" × 0 for all "a" in "A" (which is proved below). This is true for any context in which addition and multiplication have group structures defined on the same set (for example, an algebra). Property 2 is, however, "not" a natural consequence of this interplay, as there are algebraic structures in which addition and multiplication are defined which do not have the zero-product property.
The zero-product property
The formal statement of the zero-product property is:
* Let "A" be a ring, and let "B" be a
subset of "A". The "zero-product property" holds in "B" if for all elements "a" and "b" in "B" we have "ab" = 0 implies "a" = 0 or "b" = 0.Examples
* An
integral domain is a ring in which, by definition, the zero-product property holds. Thus, for example, the zero-product property holds in the ring ofinteger smodulo "p",::Z"p" = {0, 1, 2, ..., "p" − 1}
:where "p" is a prime, the
integer s Z, therational number s Q, thereal number s R, and thecomplex number s C since these are all integral domains.* Since any field is an integral domain, the zero-product property holds in any field.
* In the
skew field ofquaternions , the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative.* Any
commutative ring with the zero-product property is "almost" anintegral domain (because it may not have unity); this deficiency is not critical, because the construction of thefield of fractions works for this ring.Examples of structures which do not possess the zero-product property
It is not, however, true that the zero-product property necessarily holds in every structure which possesses an additive and multiplicative operation.
* If a commutative ring "R" is not an
integral domain , then the zero-product property does not hold: that is, there are non-zero "a" and "b" in "R" such that "ab" = 0. For a specific example of this, if "R" is the ring ofinteger smodulo 6, or Z6 = {0, 1, 2, 3, 4, 5} then there are 2, 3 in Z6 such that 2 × 3 ≡ 0 (mod 6) but neither 2 ≡ 0 (mod 6) nor 3 ≡ 0 (mod 6).
* In general, Z"n", where "n" is acomposite number , is not anintegral domain and therefore the zero-product property does not hold. For if ::"n" = "m" × "q" : with "m", "q" < "n", then neither "m" nor "q" is equal to 0 (mod "n"), but "m" × "q" = "n" = 0 (mod "n"), violating the zero-product property.
* If "R" is the ring M2(Z) of 2 by 2 matrices withinteger coefficient s then there are matrices:: and
:neither of which is equal to the zero matrix
::
:such that
::,
:so M2(Z) does not possess the zero-product property.
* The ring of all functions "f" : [0, 1] → R from the
unit interval to thereal number s does not possess the zero-product property (that is, is not anintegral domain ) because there are functions which are not identically equal to zero yet for which their product in the ring is the zero function.Zero divisors
In a ring "R" the set of elements for which the zero-product rule does "not" hold are called the
zero divisor s of the ring, denoted by "Z"("R"). Thus "Z"("R") contains every element which can be multiplied by some non-zero element to produce zero. In set notation::"Z"("R") = {"r" in "R" | there is a non-zero "s" in "R" such that "rs" = 0 or "sr" = 0}.
Thus the zero divisors of a ring "R" are a measure of how much the zero-product property holds in "R".
* 0 is a zero divisor in any ring except the zero ring.
* The zero divisors of an
integral domain consist only of the element 0, or "Z"("R") = {0}.Uses in analysis
The zero-product property is used in solving
polynomial equations over thereal number s, such asquadratic equation s.* For example, when finding all values of "x" which satisfy "x"2 + "x" − 6 = 0, we first factor the left side of the equation to obtain ("x" + 3)("x" − 2) = 0. Then, by the zero-product property, we know that either "x" + 3 = 0 (in which case "x" = − 3), or "x" − 2 = 0 (in which case "x" = 2). Thus, our solution is all "x" in the set {−3, 2}.
* This method can be extended to polynomials of higher degree. In general, if a polynomial of degree "n" with coefficients in a ring "R" can be written as a product of factors ("x" − "a"1)("x" − "a"2) ... ("x" − "a""n") = 0 then thesolution set is {"a"1, "a"2, ..., "a""n"}.* In the
complex number s the extension given above applies to "any" polynomial. Thus a polynomial "f"("z") of degree "n" over C can be written as a product of "n" factors ("z" − α) where α is one of precisely "n" roots of "f". This is known as thefundamental theorem of algebra .Note that solving quadratic equations in algebraic structures in which the zero-product property does not hold can lead to surprising results. For example, the quadratic equation
:"x"2 − "x" = 0
has solutions {0, 1} in Z, Q, or R, but in Z6 the
solution set is {0, 1, 3, 4} since 32 − 3 = 6 ≡ 0 (mod 6) and 42 − 4 = 12 ≡ 0 (mod 6).Proofs
="a" × 0 = 0=Let "A" be ring with more than one element and let "a" be a non-zero element of "A". Then "a" × 0 = "a" × ("a" + −"a") = "a"2 + (−"a"2) = 0.
ee also
*
Fundamental theorem of algebra
*Integral domain
*Prime ideal
*Zero divisor sExternal links
* [http://planetmath.org/encyclopedia/ZeroRuleOfProduct.html PlanetMath: Zero rule of product]
References
David S. Dummit, Richard M. Foote, "Abstract Algebra", Wiley (3d ed.): 2003, ISBN 0-471-43334-9.
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