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Topic: Semiprime

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	PlanetMath: semiprime ideal
		itself satisfies all of these conditions (including being expressed as an intersection of an empty family of prime ideals) and is thus semiprime.
		is said to be a semiprime ring if its zero ideal is a semiprime ideal.
		This is version 7 of semiprime ideal, born on 2001-11-23, modified 2003-11-29.
	planetmath.org /encyclopedia/SemiprimeIdeal.html (119 words)

	Semiprime Information
		In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers.
		This is the square of the largest known prime number; the square of any prime number is semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known.
		Semiprimes are highly useful in the area of cryptography and number theory, most notably in public key cryptography, where they are used by RSA and pseudo-random number generators such as Blum Blum Shub.
	www.bookrags.com /wiki/Semiprime (346 words)

	NationMaster - Encyclopedia: Semiprime
		This is the square of the largest known prime number; the square of any prime number is semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known.
		Semiprimes are highly useful in the area of cryptography and number theory, most notably in public key cryptography, where they are used by RSA and pseudo-random number generators such as Blum Blum Shub.
		The value of Euler's totient function for a semiprime n = pq is particularly simple when p and q are distinct: The first thousand values of Ï†(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n.
	www.nationmaster.com /encyclopedia/Semiprime (1070 words)

	DC MetaData for:A classification of semiprime segments
		A semiprime segment of a ring $R$ is a pair $P_2 \subset P_1$ of semiprime ideals of $R$ such that $\bigcap I^n \subseteq P_2$ for all ideals $I$ of $R$ with $P_2 \subset I \subset P_1$.
		In this paper semiprime segments with $P_1$ a comparizer ideal are classified as either simple, or exceptional, or archimedean, extending to several classes of rings a classification known for right chain rings.
		These three types of semiprime segments are also characterized in terms of the pseudo-radical.
	www.uni-duisburg.de /FB11/PUBL/SHADOW/ToernSgments.rdf.html (103 words)

	Semiprime Ideals
		Review the earlier proof, and apply it to AA instead of AB. The ideal C is semiprime iff xRx in C implies x is in C, for all x in R. The homomorphic image of a semiprime ideal is semiprime.
		The intersection of semiprime ideals is semiprime; A
		Once again the minimal semiprime ideal beneath a given semiprime ideal is always 0 in a domain.
	www.mathreference.com /ring,semi.html (166 words)

	PlanetMath: semiprime ideal
		itself satisfies all of these conditions (including being expressed as an intersection of an empty family of prime ideals) and is thus semiprime.
		is said to be a semiprime ring if its zero ideal is a semiprime ideal.
		This is version 7 of semiprime ideal, born on 2001-11-23, modified 2003-11-29.
	www.planetmath.org /encyclopedia/SemiprimeIdeal.html (120 words)

	Informat.io on Semiprime (Site not responding. Last check: 2007-11-05)
		This is the square of the largest known prime number; the square of any prime number is semiprime, so the largest known semiprime will always be the square of the largest known prime.
		Semiprimes are highly useful in the area of cryptography and number theory, most notably in public key cryptography, where it is used by RSA and pseudo-random number generators such as Blum Blum Shub.
		In practical cryptography, it is not sufficient to choose just any semiprime; a good number must evade a number of well-known special-purpose algorithms that can factor numbers of certain form.
	www.informat.io /?title=semiprime-1 (0 words)

	NationMaster - Encyclopedia: Twin prime conjecture
		The approach he took involved a topic called sieve theory, and he managed to treat the twin prime conjecture and Goldbach's conjecture in similar manners.
		Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics.
		Defining a Chen prime to be a prime p such that p + 2 is either a prime or a semiprime, Terence Tao and Ben Green showed in 2005 that there are infinitely many three term arithmetic progressions of Chen primes.
	www.nationmaster.com /encyclopedia/Twin-prime-conjecture (1287 words)

	Semiprime - Definition, explanation
		Semiprimes are highly useful in the area of cryptography and number theory, the most notable examples of which being public key cryptography (ie.
		The main rationale for the use of such numbers is the inherent difficulty (at least presently) of integer factorization, especially as the size of the numbers increase.
		Put simply, it's rather simple to see that 35 can be factored into the product of 5 and 7, but if you can factor this large semiprime, you win $200,000.
	www.calsky.com /lexikon/en/txt/s/se/semiprime.php (180 words)

	Semiprime (Site not responding. Last check: 2007-11-05)
		Semiprimes are highly useful in the area of cryptography and number theory, most notably in public key cryptography (as used by RSA) and pseudo-random number generators (an example being Blum Blum Shub).
		In practical cryptography, it's not sufficient to choose just any semiprime; a good number must evade a number of well-known special-purpose algorithms that can factor numbers of certain form.
		The value of Euler's totient function for a semiprime n = pq is particularly simple when p and q are distinct:
	www.abcworld.net /Semiprime.html (319 words)

	Semiprime Ideal -- from Wolfram MathWorld
		A prime ideal is certainly semiprime, but the latter is a strictly more general notion.
		is a semiprime number, which causes a conflict in terminology.
		In general, the semiprime ideals of a principal ideal domain are the proper ideals whose generator has no multiple prime factors.
	mathworld.wolfram.com /SemiprimeIdeal.html (0 words)

	loan Semiprime - loan-reports.com (Site not responding. Last check: 2007-11-05)
		Semiprimes are highly useful in the area of cryptography and number theory, most notably in public key cryptography, where they are used by RSA and pseudorandom number generators such as Blum Blum Shub.
		The factors p and q of n should be very large, around the same order of magnitude as the square root; this makes trial division and Pollard's rho algorithm impractical.
		The number 1679 = 23Ã—73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns.
	www.loan-reports.com /Semiprime (556 words)

	Ladislav Beran (Site not responding. Last check: 2007-11-05)
		The semiprimeness is also used to characterize distributive and modular lattices.
		Prime ideals are described as the meet-irreducible semiprime ideals.
		In addition, a necessary and sufficient condition for the equality of prime radicals is obtained.
	mathnet.preprints.org /EMIS/journals/CMUC/cmuc9404/abs/beran.htm (90 words)

	The On-Line Encyclopedia of Integer Sequences
		a(0) = 4 because 5^0 + 3 = 4 = A001358(1), and no semiprime is closer to 5^0.
		a(1) = 4 because 5^1 - 1 = 4 = A001358(1), and no semiprime is closer to 5^1.
		a(2) = 25 because 5^2 + 0 = 25 = A001358(9), no semiprime is closer to 5^2.
	www.research.att.com /~njas/sequences/A117429 (271 words)

	Amazon.com Books: Semiprime (Site not responding. Last check: 2007-11-05)
		Localization of non-commutative noetherian rings at semiprime ideals: Vorlesungsausarb (Algebra-Berichte - Mathematisches Institut der UniversitaÌt MuÌnchen) (Algebra-Berichte - Mathematisches Institut der UniversitaÌt MuÌnchen) by Bruno J MuÌller (Unknown Binding - Jan 1, 1974)
		It includes a char- acterization of left Noetherian semiprime rings (that is, left Noetherian rings with zero prime radical).
		The next definition is similar to the usual definition of semiprime and prime ideals, but instead of ordinary ideals, we consider...
	www.amazon.com /s?ie=UTF8&keywords=Semiprime&tag=httpexplaguid-20&index=books&link_code=qs&page=1 (0 words)

	Semiprime and DCC (Site not responding. Last check: 2007-11-05)
		A division ring is semiprime, with 0 being a semiprime ideal.
		By correspondence, 0 is a semiprime ideal in the matrix ring, creating a larger semiprime ring.
		A jacobson semisimple ring has a lower nil radical of 0, whence 0 is a semiprime ideal, and the ring is semiprime.
	www.mathreference.com /ring-rad,spdcc.html (422 words)

	Conjecture 49. n = pq +rs
		If we define an 'Extended Semiprime' as a semiprime whose each factor is
		Every Extended Semiprime > 1 is the sum of two Extended Semiprimes.
		A semiprime is even iff at least one of the two prime factors is equal to 2.
	www.primepuzzles.net /conjectures/conj_049.htm (1509 words)

	What is PQ? - A Forum Post by HsojVvad (Site not responding. Last check: 2007-11-05)
		Currently, the largest known semiprime is (232,582,657 − 1)2, which has over 19 million digits.
		The factors p and q of n should be very large, around the same order of magnitude as the square root; this makes trial division and Pollard's rho algorithm impractical.
		The value of Euler's totient function for a semiprime n = pq is particularly simple when p and q are distinct:
	forums.galciv2.com /index.aspx?aid=133238&c=1 (1693 words)

	Semiprime -- from Wolfram MathWorld
		The first few semiprimes whose factors are distinct (i.e., the squarefree semiprimes) are 6, 10, 14, 15, 21, 22, 26, 33, 34,...
		The square of any prime number is by definition a semiprime.
		The largest known semiprime is therefore the square of the largest known prime.
	mathworld.wolfram.com /Semiprime.html (0 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		It is proved that every nonsingular essentially compressible module $M$ is isomorphic to a submodule of a free module, and the converse holds in case $R$ is semiprime right Goldie.
		In case $R$ is a right FBN ring, $M$ is essentially compressible if and only if $M$ is subisomorphic to a direct sum of critical compressible modules.
		It follows that a ring $R$ is semiprime right Goldie if and only if $R$ is a right essentially compressible ring with at least one uniform right ideal.
	www.ipm.ac.ir /IPM/pdf/abs8001.tex (159 words)

	Some Epimorphic Regular Contexts (ResearchIndex)
		A von Neumann regular extension of a semiprime ring naturally defines a epimorphic extension in the category of rings.
		These are studied, and four natural examples are considered, two in commutative ring theory, and two in rings of continuous functions.
		Introduction In this note we study certain extensions of commutative semiprime rings which we call epimorphic regular contexts, or sometimes simply contexts.
	citeseer.ist.psu.edu /raphael99some.html (0 words)

	UWM Math: Noetherian Rings
		This result is much deeper than Ore's 1931 result, and the techniques and subsidiary results gave researchers technical tools that are still in use today.
		[Goldie showed a noetherian ring has a semisimple ring of fractions if and only if it is semiprime and it has a division ring of fractions if and only if it has no zero divisors.]
		Here are some examples of non-commutative noetherian rings that have been studied by members of the algebra group:
	www.uwm.edu /Dept/Math/Research/Algebra/noetherian/noetherian.html (467 words)

	Semiprime, prime and simple Jordan superpairs covered by grids
		Semiprime, prime and simple Jordan superpairs covered by grids
		This paper is a continuation of the investigation of Jordan superpairs covered by a grid which were recently described by the second author ([83] on this server).
		For such a Jordan superpair we characterize semiprimeness, primeness and simplicity in terms of its associated supercoordinate system.
	www.uibk.ac.at /mathematik/loos/jordan/archive/jsupgrid/index.html (63 words)

	[No title]
		I pestered him briefly, to explain what he meant by "interesting".
		He never explained it, but all the while, I believed that I knew what he meant: An Interesting SemiPrime (ISP) is a number which - is the product of two primes; - one has a concise proof that it's the product of two primes; and - there is scant hope of finding a proper factor.
		But the semiprime proof might reveal enough about N, that it can be factored.
	www.graysage.com /djr/isp.txt (318 words)

	Mathpuzzle.com
		Atomium, which I suppose would have square tilings in its bathrooms.
		Semiprimes, numbers which are the product of two primes.
		It seems to be much harder to find interesting semiprimes.
	www.mathpuzzle.com /29Jun2003.html (3465 words)

	Welcome to Dr. Samman's Home-Page (Site not responding. Last check: 2007-11-05)
		Thaheem, A. and Samman, M. S., “A note on α-derivations on semiprime rings”, Demonstratio Mathematica, (34) 4 (2001), 783-788.
		Samman, M. and Thaheem, A. B., “Derivations on semiprime rings”, Int.J. of pure and applied Mathematics, 5(4)(2003), 469-477.
		Samman, M. and M.A. Chaudhry, “Generalized inverses of semiprime rings”, Aequationes Mathematicae, (accepted).
	faculty.kfupm.edu.sa /math/msamman/publications.htm (0 words)

	Rings Of Quotients Of f-Rings By Gabriel Filters of Ideals (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Abstract: The article examines the role of Gabriel filters of ideals in the context of semiprime f--rings.
		It is shown that for every 2--convex semiprime f--ring A and every multiplicative filter G of dense ideals the ring of quotients of A by G, namely the direct limit of the HomA(I;A) over all I 2 G, is an `--subring of QA, the maximum ring of quotients.
		Relative to the category of all commutative rings with identity, it is shown that for every 2-- convex semiprime f--ring A, qA, the classical...
	citeseer.ist.psu.edu /476597.html (0 words)

	Primitive ideals
		If is the smallest semiprime ideal of R containing I.
		Proof:Let J be the sum of the elements of
		If J' is a semiprime ideal containing I, the image of J in R/J' is nilpotent, and hence
	www.mathematik.uni-bielefeld.de /%7Ephilfahr/enveloping/node14.html (288 words)

	NSDL Metadata Record -- Chen prime
		is either a prime or a semiprime, then p is called a...
		The name was assigned by Ben Green and Terrence Tao in recognition of Chen's theorem that every sufficiently large even number can be written as the sum of a prime and a semiprime.
		To give two examples of Chen primes: 41 is a Chen prime since 43 is also a prime, but 43 is itself not a Chen prime because 45 has one factor too many to be a semiprime; 47 is a Chen prime since 49, the square of a prime, is a semiprime.
	nsdl.org /mr/2578079 (232 words)

	MathReference.com archive
		The Math Reference project is essentially a self-paced tutorial/archive, written in English/html, that takes the reader through modern mathematics using modern techniques.
		In other words, we won't stop at prime numbers, we'll cover prime elements in a commutative ring, prime ideals, semiprime ideals, and many other beautiful, abstract topics.
		Start with "number theory" from the menu of topics above, then explore other topics (breadth), or investigate numbers and modularity in more detail (depth).
	www.mathreference.com (1141 words)

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