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...

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...

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...

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...

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...

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...

denominators in I {\displaystyle I} , hence the name fractional ideal. The principal fractional ideals are those R {\displaystyle R} -submodules of K {\displaystyle...

principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals...

maximal ideals are the principal ideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principal ideal domain....

algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated...

nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This...

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...

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...

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...

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...

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...

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...

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...

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...

(algebra) Ring ideal Principal ideal Ideal quotient Maximal ideal, minimal ideal Primitive ideal, prime ideal, semiprime ideal Radical of an ideal Jacobson...

and the boys that it appeared as though the college had found its ideal principal. However, in 1875 Hartley resigned to become president of the newly...

Ideal School and College (Bengali: আইডিয়াল স্কুল অ্যান্ড কলেজ), commonly known as Motijheel Ideal School and College (despite there being 2 more branches...

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...

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...

{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 /...

{\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...

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...

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...

integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields...

This research revealed that the personality characteristics of the ideal principal are strong mindedness, directness, top-down management and charisma...