About: Principal ideal domain

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In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains appear in the following chain of class inclusions:

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rdf:type	protein
rdfs:label	Principal ideal domain (en)
rdfs:comment	In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains appear in the following chain of class inclusions: (en)
sameAs	Principal ideal domain
dbp:wikiPageUsesTemplate	dbt:! dbt:Algebraic_structures dbt:Clarify dbt:Main dbt:Math dbt:Mvar dbt:Short_description dbt:Isbn dbt:Commutative_ring_classes
Subject	Commutative algebra Ring theory
gold:hypernym	Domain
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Principal_ideal_domain?oldid=1018309235&ns=0
Wikipage page ID	25039 (xsd:integer)
page length (characters) of wiki page	9709 (xsd:nonNegativeInteger)
Wikipage revision ID	1018309235 (xsd:integer)
has abstract	In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax + by. Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains. Principal ideal domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Principal_ideal_domain
is Wikipage redirect of	Principal ring Principle ideal domain
is Link from a Wikipage to another Wikipage of	Greatest common divisor Local ring Nagata ring Noetherian ring Novikov ring Abelian group Affine variety Ascending chain condition on principal ideals Basic subgroup Bézout's identity Bézout domain Canonical form Category of rings Free abelian group Free ideal ring Global dimension Glossary of commutative algebra Glossary of ring theory Jacobi operator Jacobson's conjecture Jordan normal form Discrete valuation ring Domain Chinese remainder theorem Cycle decomposition Dedekind domain Dedekind–Hasse norm Degree of a polynomial Linear algebra Linear equation over a ring Linear relation List of number fields with class number one Cofree coalgebra Commutative ring Hensel's lemma Cubic function List of commutative algebra topics Minimal polynomial (field theory) Minimal polynomial (linear algebra) Minimal prime ideal Elementary divisors Finitely generated module Flat module Functional calculus Fundamental theorem of arithmetic GCD domain Gauss's lemma (polynomials) Gaussian integer Ideal (ring theory) Ideal class group Indecomposable module Factorization Thue's lemma Group cohomology Injective module Integer Integrally closed domain Invariant factor Irreducible ideal Krull dimension Künneth theorem Modular representation theory Module (mathematics) Euclidean algorithm Euclidean domain Least common multiple Matrix equivalence Matrix similarity Maximal ideal Multiplicative function
is known for of	Theodore Motzkin

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