[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/zero-product-property-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki40\/zero-product-property-wikipedia\/","headline":"Zero-product property – Wikipedia","name":"Zero-product property – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In algebra, the zero-product property states that the product of two nonzero elements","datePublished":"2014-02-11","dateModified":"2014-02-11","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki40\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8fe3b89490f9893320c6551017c72a8319d01c48","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8fe3b89490f9893320c6551017c72a8319d01c48","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki40\/zero-product-property-wikipedia\/","wordCount":8920,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, if\u00a0ab=0,\u00a0then\u00a0a=0\u00a0or\u00a0b=0.{displaystyle {text{if }}ab=0,{text{ then }}a=0{text{ or }}b=0.}       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4This 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 \u2014 the integers Z{displaystyle mathbb {Z} }, the rational numbers Q{displaystyle mathbb {Q} }, the real numbers R{displaystyle mathbb {R} }, and the complex numbers        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4C{displaystyle mathbb {C} } \u2014 satisfy the zero-product property.  In general, a ring which satisfies the zero-product property is called a domain.Table of ContentsAlgebraic context[edit]Examples[edit]Non-examples[edit]Application to finding roots of polynomials[edit]See also[edit]References[edit]External links[edit]Algebraic context[edit]Suppose A{displaystyle A} is an algebraic structure.  We might ask, does A{displaystyle A} have the zero-product property?  In order for this question to have meaning, A{displaystyle A} must have both additive structure and multiplicative structure.[2]  Usually one assumes that A{displaystyle A} is a ring, though it could be something else, e.g. the set of nonnegative integers {0,1,2,\u2026}{displaystyle {0,1,2,ldots }} with ordinary addition and multiplication, which is only a (commutative) semiring.Note that if A{displaystyle A} satisfies the zero-product property, and if B{displaystyle B} is a subset of A{displaystyle A}, then B{displaystyle B} also satisfies the zero product property: if a{displaystyle a} and b{displaystyle b} are elements of B{displaystyle B} such that ab=0{displaystyle ab=0}, then either a=0{displaystyle a=0} or b=0{displaystyle b=0} because a{displaystyle a} and b{displaystyle b} can also be considered as elements of A{displaystyle 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.If p{displaystyle p} is a prime number, then the ring of integers modulo p{displaystyle 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 {0,1,2,\u2026}{displaystyle {0,1,2,ldots }} is not a ring (being instead a semiring), but it does satisfy the zero-product property.Non-examples[edit]Let Zn{displaystyle mathbb {Z} _{n}} denote the ring of integers modulo n{displaystyle n}. Then Z6{displaystyle mathbb {Z} _{6}} does not satisfy the zero product property: 2 and 3 are nonzero elements, yet 2\u22c53\u22610(mod6){displaystyle 2cdot 3equiv 0{pmod {6}}}.In general, if n{displaystyle n} is a composite number, then Zn{displaystyle mathbb {Z} _{n}} does not satisfy the zero-product property.  Namely, if n=qm{displaystyle n=qm} where 02{displaystyle mathbb {Z} ^{2times 2}} of 2\u00d72 matrices with integer entries does not satisfy the zero-product property: if M=(1\u2212100){displaystyle M={begin{pmatrix}1&-1\\0&0end{pmatrix}}} and N=(0101),{displaystyle N={begin{pmatrix}0&1\\0&1end{pmatrix}},} then MN=(1\u2212100)(0101)=(0000)=0,{displaystyle MN={begin{pmatrix}1&-1\\0&0end{pmatrix}}{begin{pmatrix}0&1\\0&1end{pmatrix}}={begin{pmatrix}0&0\\0&0end{pmatrix}}=0,} yet neither M{displaystyle M} nor N{displaystyle N} is zero.The ring of all functions f:[0,1]\u2192R{displaystyle 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 \u2265 2, functions f1,\u2026,fn{displaystyle f_{1},ldots ,f_{n}}, none of which is identically zero, such that fifj{displaystyle f_{i},f_{j}} is identically zero whenever i\u2260j{displaystyle ineq j}.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[edit]Suppose P{displaystyle P} and Q{displaystyle Q} are univariate polynomials with real coefficients, and x{displaystyle x} is a real number such that P(x)Q(x)=0{displaystyle P(x)Q(x)=0}.  (Actually, we may allow the coefficients and x{displaystyle x} to come from any integral domain.)  By the zero-product property, it follows that either P(x)=0{displaystyle P(x)=0} or Q(x)=0{displaystyle Q(x)=0}.  In other words, the roots of PQ{displaystyle PQ} are precisely the roots of P{displaystyle P} together with the roots of Q{displaystyle Q}.Thus, one can use factorization to find the roots of a polynomial.  For example, the polynomial x3\u22122x2\u22125x+6{displaystyle x^{3}-2x^{2}-5x+6} factorizes as (x\u22123)(x\u22121)(x+2){displaystyle (x-3)(x-1)(x+2)}; hence, its roots are precisely 3, 1, and \u22122.In general, suppose R{displaystyle R} is an integral domain and f{displaystyle f} is a monic univariate polynomial of degree d\u22651{displaystyle dgeq 1} with coefficients in R{displaystyle R}.  Suppose also that f{displaystyle f} has d{displaystyle d} distinct roots r1,\u2026,rd\u2208R{displaystyle r_{1},ldots ,r_{d}in R}.  It follows (but we do not prove here) that f{displaystyle f} factorizes as f(x)=(x\u2212r1)\u22ef(x\u2212rd){displaystyle f(x)=(x-r_{1})cdots (x-r_{d})}.  By the zero-product property, it follows that r1,\u2026,rd{displaystyle r_{1},ldots ,r_{d}} are the only roots of f{displaystyle f}: any root of f{displaystyle f} must be a root of (x\u2212ri){displaystyle (x-r_{i})} for some i{displaystyle i}.  In particular, f{displaystyle f} has at most d{displaystyle d} distinct roots.If however R{displaystyle R} is not an integral domain, then the conclusion need not hold.  For example, the cubic polynomial x3+3x2+2x{displaystyle x^{3}+3x^{2}+2x} has six roots in Z6{displaystyle mathbb {Z} _{6}} (though it has only three roots in Z{displaystyle mathbb {Z} }).See also[edit]^ The other being a\u22c50 = 0\u22c5a = 0.  Mustafa A. Munem and David J. Foulis, Algebra and Trigonometry with Applications (New York:  Worth Publishers, 1982), p. 4.^ 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\u00a00-471-43334-9.External links[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/zero-product-property-wikipedia\/#breadcrumbitem","name":"Zero-product property – Wikipedia"}}]}]