PlanetMath: PID

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

PID

(Definition)

A principal ideal domain is an integral domain where every ideal is a principal ideal.

In a PID, an ideal $(p)$ is maximal if and only if $p$ is irreducible (and prime since any PID is also a UFD).

Note that subrings of PIDs are not necessarily PIDs. (There is an example of this within the entry biquadratic field.)

Anyone with an account can edit this entry. Please help improve it!

"PID" is owned by mps. [ full author list (4) | owner history (2) ]

(view preamble | get metadata)

Other names:

principal ideal domain

Attachments:: Dedekind-Hasse valuation (Definition) by Henry
example of PID (Example) by sleske
every PID is a UFD (Theorem) by rm50
PID and UFD are equivalent in a Dedekind domain (Theorem) by rm50

Log in to rate this entry.
(view current ratings)

Cross-references: biquadratic field, subrings, prime, irreducible, principal ideal, ideal, integral domain
There are 43 references to this entry.

This is version 8 of PID, born on 2001-11-04, modified 2007-05-27.
Object id is 674, canonical name is PID.
Accessed 10857 times total.

Classification:

AMS MSC:	13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)
	11N80 (Number theory :: Multiplicative number theory :: Generalized primes and integers)
	13G05 (Commutative rings and algebras :: Integral domains)
	16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)

Pending Errata and Addenda

None.[ View all 5 ]

Discussion

forum policy

Interact