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...

every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID) is an integral domain in which every ideal is principal...

are studied in domains as Bézout domains. A principal ideal ring which is also an integral domain is said to be a principal ideal domain (PID). In this...

concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional...

Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or...

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

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

In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is also a principal ideal. This means that Bézout's identity...

definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if...

not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is...

⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ euclidean domains ⊃ fields ⊃...

⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ euclidean domains ⊃ fields ⊃...

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 that satisfies any...

modules over principal ideal domains (PIDs) are classified by the structure theorem for finitely generated modules over a principal ideal domain: the primary...

system for details. More generally, linear algebra is effective on a principal ideal domain if there are algorithms for addition, subtraction and multiplication...

(commutative) principal ideal domain and M is a finitely generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain...

A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem...

congruences) is true over every principal ideal domain. It has been generalized to any ring, with a formulation involving two-sided ideals. The earliest known statement...

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...

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 Krull...

g. principal ideal domains) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notably unique factorization domains and...

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

⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ euclidean domains ⊃ fields ⊃...

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...

The next simplest case is the case when the coefficient ring is a principal ideal domain. This case is particularly important because the integers Z {\displaystyle...

let R be a principal ideal domain, K be its field of fractions, and π be an irreducible element of R. Since every principal ideal domain is a unique...

rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers...

domain into a principal ideal domain. To wit, they proved that if an integral domain R has a Dedekind–Hasse norm, then R is a principal ideal domain....

defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be...

possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector...