A Hasse diagram of a portion of the lattice of ideals of the integers Z. The purple and green nodes indicate semiprime ideals. The purple nodes are prime ideals, and the purple and blue nodes are primary ideals.
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as reduced rings.
For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form where n is a square-free integer. So, is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but is not (because 12 = 22 × 3, with a repeated prime factor).
The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings.
Most definitions and assertions in this article appear in (Lam 1999) and (Lam 2001).
In ring theory, a branch of mathematics, semiprime ideals and semiprimerings are generalizations of prime ideals and prime rings. In commutative algebra...
generalized to non-commutative rings in the semiprimering article. The radical of an ideal I {\displaystyle I} in a commutative ring R {\displaystyle R} , denoted...
I of a ring R is semiprime if for any ideal A of R, An ⊆ I implies A ⊆ I. Equivalently, I is semiprime if and only if R/I is a semiprimering. semiprimitive...
the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients...
ideals are prime, and prime ideals are both primary and semiprime. An ideal P of a commutative ring R is prime if it has the following two properties: If...
Primitive ring, Semiprimitive ring Prime ring, Semiprimering, Reduced ring Integral domain, Domain (ring theory) Field of fractions, Integral closure Euclidean...
determines the structure of primitive rings Goldie's theorem determines the structure of semiprime Goldie rings The Zariski–Samuel theorem determines...
minimal ideals of a ring R, then N1N2 = {0}. A simple ring with a minimal right ideal is a semisimple ring. In a semiprimering, there exists a minimal...
numeric names: authors list (link) Martindale, Wallace S. (1973). "On semiprime P. I. Rings". Proceedings of the American Mathematical Society. 40 (2): 365...
Jacobson ring. The Baer radical of a ring is the intersection of the prime ideals of the ring R. Equivalently it is the smallest semiprime ideal in R...
must be greater than 1400. Almost prime Fermi–Dirac prime Perfect power Semiprime Crandall, Richard; Pomerance, Carl B. (2005). Prime Numbers: A Computational...
(mod N). The structure of the RSA public key requires that N be a large semiprime (i.e., a product of two large prime numbers), that 2 < e < N, that e be...
assume N = pq is a semiprime then the following process can be used to compute an eta-quotient basis of Mk(Γ0(N)). Fix a semiprime N = pq which is coprime...
Fundamental theorem of arithmetic Factorization forms Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular...
sufficiently large even number can be expressed as the sum of a prime and a semiprime (the product of two primes). Also, any even integer greater than 10 can...
primary, and moreover an ideal is prime if and only if it is primary and semiprime (also called radical ideal in the commutative case). Every primary ideal...
and include principal ideal domains, Euclidean domains, and polynomial rings over a field. However, the theorem does not hold for algebraic integers...
ISBN 978-0-521-82356-2, MR 2131318; see especially Lemma 8.2 (Ring Lemma), pp. 73–74, and Appendix B, The Ring Lemma, pp. 318–321. Knuth, Donald E (1997), The Art...
third power of two (23). 8 is the first number which is neither prime nor semiprime and the only nonzero perfect power that is one less than another perfect...
first prime and composite, 2 and 4. 6 is a pronic number and the only semiprime to be. It is the first discrete biprime (2 × 3) which makes it the first...
the 3-aliquot tree. It is the first member of the first cluster of two semiprimes (9, 10), preceding (14, 15). Casting out nines is a quick way of testing...
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