In mathematics, specifically ring theory, a principal ideal is an ideal I {\displaystyle I} in a ring R {\displaystyle R} that is generated by a single...
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principal ideal domain, or PID, is an integral domain (that is, a commutative ring without nonzero zero divisors) in which every ideal is principal (that...
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algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian...
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Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples...
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In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The...
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In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the...
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denominators in I {\displaystyle I} , hence the name fractional ideal. The principal fractional ideals are those R {\displaystyle R} -submodules of K {\displaystyle...
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algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated...
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principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals...
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fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization...
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maximal ideals are the principal ideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principal ideal domain....
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nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This...
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a prime ideal. Given an integral domain R {\displaystyle R} , any prime element p ∈ R {\displaystyle p\in R} generates a principal prime ideal ( p ) {\displaystyle...
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smallest ideal that contains a given element p is a principal ideal and p is said to be a principal element of the ideal in this situation. The principal ideal...
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prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem...
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Cardinal point (optics) (redirect from Principal plane)
system. These are the focal points, the principal points, and the nodal points; there are two of each. For ideal systems, the basic imaging properties such...
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(algebra) Ring ideal Principal ideal Ideal quotient Maximal ideal, minimal ideal Primitive ideal, prime ideal, semiprime ideal Radical of an ideal Jacobson...
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Krull dimension (redirect from Height of an ideal)
dimension 0; more generally, k[x1, ..., xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1. A local ring has...
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algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain R...
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partial order (P, ≤) is a meet prime element when p is the principal element of a principal prime ideal. Equivalently, if P is a lattice, p ≠ top, and for all...
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Commutative ring (section Principal ideal domains)
{\displaystyle s} . Such an ideal is called a principal ideal. If every ideal is a principal ideal, R {\displaystyle R} is called a principal ideal ring; two important...
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fractional ideals form a group under a product. The quotient of the group of fractional ideals by the subgroup of principal ideals is then the ideal class...
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{a}}^{n}} , where n = [ L : K ] {\displaystyle n=[L:K]} . The ideal norm of a principal ideal is thus compatible with the field norm of an element: N B /...
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The thin ideal is the concept of the ideally slim female body. The common perception of this ideal is a woman who possesses a slender, feminine physique...
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primary decomposition. Every primary ideal of a principal ideal domain is an irreducible ideal. Every irreducible ideal is primal. An element of an integral...
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Ideal School and College (Bengali: আইডিয়াল স্কুল অ্যান্ড কলেজ), commonly known as Motijheel Ideal School and College (despite there being 2 more branches...
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integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields...
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Bézout's identity (section For principal ideal domains)
Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in all principal ideal domains. If a and b...
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right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero left ideal of...
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Algebraic number theory (section Ideal class group)
principal ideal theorem, every prime ideal of O generates a principal ideal of the ring of integers of E. A generator of this principal ideal is called...
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