en>Hhhippo: Removing redundant category: Physics

2012-09-15T21:28:10Z

Removing redundant category: Physics

← Older revision		Revision as of 23:28, 15 September 2012
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	~~psychic readers (~~[~~http~~://~~bigpolis~~.~~com~~/~~blogs~~/~~post~~/~~6503 bigpolis~~.~~com~~]) ~~Hi there~~. ~~Allow me begin by introducing~~ the ~~writer~~, ~~her title~~ is ~~Sophia~~. ~~What me~~ and ~~my family love~~ is ~~bungee leaping but I've been taking on new issues lately~~. ~~My working day job~~ is an ~~invoicing officer~~ but ~~I've~~ ~~[http~~://~~findyourflirt~~.~~net~~/~~index~~.~~php?m~~=~~member_profile~~&p=~~profile~~&id=~~117823 certified psychics~~] ~~already applied~~ for ~~an additional~~ one. For ~~many years she's been living~~ in ~~Kentucky~~ but ~~her spouse desires them to transfer~~.<br><br>~~Here is my homepage~~ :~~: email psychic readings~~ (~~[http:~~//~~203~~.~~250~~.78.~~160~~/~~zbxe~~/~~?document_srl~~=~~1792908 you could look here~~])		In [[algebra]], the '''zero-product property''' states that the product of two nonzero elements is nonzero. In other words, it is the following assertion:
			<blockquote>
			If <math>ab = 0</math>, then <math>a=0</math> or <math>b=0</math>.
			</blockquote>
			The zero-product property is also known as the '''rule of zero product''' or '''nonexistence of [[zero divisor]]s.''' All of the number systems studied in [[elementary mathematics]] — the [[integer]]s <math>\mathbb{Z}</math>, the [[rational number]]s <math>\mathbb{Q}</math>, the [[real number]]s <math>\mathbb{R}</math>, and the [[complex number]]s <math>\mathbb{C}</math> — satisfy the zero-product property. In general, a [[Ring (mathematics)\|ring]] which satisfies the zero-product property is called a [[Domain (ring theory)\|domain]].

			==Algebraic context==

			Suppose <math>A</math> is an algebraic structure. We might ask, does <math>A</math> have the zero-product property? In order for this question to have meaning, <math>A</math> must have both additive structure and multiplicative structure.<ref group=note>There must be a notion of zero (the [[additive identity]]) and a notion of products, i.e., multiplication.</ref> Usually one assumes that <math>A</math> is a [[Ring (mathematics)\|ring]], though it could be something else, e.g., the nonnegative integers <math>\{0,1,2,\ldots\}</math>.

			Note that if <math>A</math> satisfies the zero-product property, and if <math>B</math> is a subset of <math>A</math>, then <math>B</math> also satisfies the zero product property: if <math>a</math> and <math>b</math> are elements of <math>B</math> such that <math>ab=0</math>, then either <math>a=0</math> or <math>b=0</math> because <math>a</math> and <math>b</math> can also be considered as elements of <math>A</math>.

			==Examples==
			* A ring in which the zero-product property holds is called a [[Domain (ring theory)\|domain]]. A [[Commutative ring\|commutative]] domain with a [[Unit element\|multiplicative identity]] element is called an [[integral domain]]. Any [[Field (abstract algebra)\|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 <math>p</math> is a [[prime number]], then the ring of [[Modular arithmetic\|integers modulo <math>p</math>]] 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 [[Skew field\|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 <math>\{0,1,2,\ldots\}</math> is not a ring, but it does satisfy the zero-product property.

			==Non-examples==

			* Let <math>\mathbb{Z}_n</math> denote the ring of [[Modular arithmetic\|integers modulo <math>n</math>]]. Then <math>\mathbb{Z}_6</math> does not satisfy the zero product property: 2 and 3 are nonzero elements, yet <math>2 \cdot 3 \equiv 0 \pmod{6}</math>.

			* In general, if <math>n</math> is a [[composite number]], then <math>\mathbb{Z}_n</math> does not satisfy the zero-product property. Namely, if <math>n = qm</math> where <math>0 < q,m < n</math>, then <math>m</math> and <math>q</math> are nonzero modulo <math>n</math>, yet <math>qm \equiv 0 \pmod{n}</math>.

			* The ring <math>\mathbb{Z}^{2 \times 2}</math> of 2 by 2 [[matrix (mathematics)\|matrices]] with [[integer]] entries does not satisfy the zero-product property: if

			::<math>M = \begin{pmatrix}1 & -1 \\ 0 & 0\end{pmatrix}</math> and <math>N = \begin{pmatrix}0 & 1 \\ 0 & 1\end{pmatrix}</math>,

			:then

			::<math>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</math>,

			:yet neither <math>M</math> nor <math>N</math> is zero.

			* The ring of all [[function (mathematics)\|function]]s <math>f: [0,1] \to \mathbb{R}</math>, from the [[unit interval]] to the [[real number]]s, has 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 <math>f_1,\ldots,f_n</math>, none of which is identically zero, such that <math>f_i \, f_j</math> is identically zero whenever <math>i \neq j</math>.

			* The same is true even if we consider only continuous functions, or only even infinitely smooth functions.

			==Application to finding roots of polynomials==
			Suppose <math>P</math> and <math>Q</math> are univariate polynomials with real coefficients, and <math>x</math> is a real number such that <math>P(x)Q(x) = 0</math>. (Actually, we may allow the coefficients and <math>x</math> to come from any integral domain.) By the zero-product property, it follows that either <math>P(x) = 0</math> or <math>Q(x) = 0</math>. In other words, the roots of <math>PQ</math> are precisely the roots of <math>P</math> together with the roots of <math>Q</math>.

			Thus, one can use [[factorization of polynomials\|factorization]] to find the roots of a polynomial. For example, the polynomial <math>x^3 - 2x^2 - 5x + 6</math> factorizes as <math>(x-3)(x-1)(x+2)</math>; hence, its roots are precisely 3, 1, and -2.

			In general, suppose <math>R</math> is an integral domain and <math>f</math> is a [[Monic polynomial\|monic]] univariate polynomial of degree <math>d \geq 1</math> with coefficients in <math>R</math>. Suppose also that <math>f</math> has <math>d</math> distinct roots <math>r_1,\ldots,r_d \in R</math>. It follows (but we do not prove here) that <math>f</math> factorizes as <math>f(x) = (x-r_1) \cdots (x-r_d)</math>. By the zero-product property, it follows that <math>r_1,\ldots,r_d</math> are the ''only'' roots of <math>f</math>: any root of <math>f</math> must be a root of <math>(x-r_i)</math> for some <math>i</math>. In particular, <math>f</math> has at most <math>d</math> distinct roots.

			If however <math>R</math> is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial <math>x^3 + 3x^2 + 2x</math> has six roots in <math>\mathbb{Z}_6</math> (though it has only three roots in <math>\mathbb{Z}</math>).

			==See also==
			* [[Fundamental theorem of algebra]]
			* [[Integral domain]] and [[domain (ring theory)\|domain]]
			* [[Prime ideal]]
			* [[Zero divisor]]

			==Notes==
			{{Reflist\|group=note}}

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

			==External links==
			* [http://planetmath.org/encyclopedia/ZeroRuleOfProduct.html PlanetMath: Zero rule of product]

			[[Category:Abstract algebra]]
			[[Category:Elementary algebra]]
			[[Category:Real analysis]]
			[[Category:Ring theory]]
			[[Category:Zero]]

en>Koumz at 23:35, 1 November 2011

2011-11-01T23:35:28Z

New page

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en>Hhhippo: Removing redundant category: Physics

en>Koumz at 23:35, 1 November 2011