View source for Principal ideal domain

{{Short description|Algebraic structure}}
{{for|a similar structure that is not necessarily commutative and may have zero-divisors|Principal ideal ring}}
In [[mathematics]], a '''principal ideal domain''', or '''PID''', is an [[integral domain]] (that is, a [[commutative ring]] without nonzero [[zero divisor]]s) in which every [[ideal (ring theory)|ideal]] is [[principal ideal|principal]] (that is, is formed by the multiples of a single element). Some authors such as [[Nicolas Bourbaki|Bourbaki]] refer to PIDs as '''principal rings'''.

Principal ideal domains are mathematical objects that behave like the [[integer]]s, with respect to [[Integral domain#Divisibility, prime and irreducible elements|divisibility]]: any element of a PID has a unique factorization into [[prime element]]s (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 {{math|''x''}} and {{math|''y''}} are elements of a PID without common divisors, then every element of the PID can be written in the form {{math|''ax'' + ''by''}}, etc.

Principal ideal domains are [[noetherian ring|Noetherian]], they are  [[integrally closed domain|integrally closed]], they are [[unique factorization domain]]s and [[Dedekind domain]]s.  All [[Euclidean domain]]s and all [[field (mathematics)|fields]] are principal ideal domains.

Principal ideal domains appear in the following chain of [[subclass (set theory)|class inclusions]]:
{{Commutative ring classes}}
{{Algebraic structures |Ring}}

==Examples==
Examples include:
* <math>K</math>: any [[field (mathematics)|field]],
* <math>\mathbb{Z}</math>: the [[ring (mathematics)|ring]] of [[integer]]s,<ref>See Fraleigh & Katz (1967), p. 73, Corollary of Theorem 1.7, and notes at p. 369, after the corollary of Theorem 7.2</ref>
* <math>K[x]</math>: [[polynomial ring|rings of polynomials]] in one variable with coefficients in a field. (The converse is also true, i.e. if <math>A[x]</math> is a PID then <math>A</math> is a field.) Furthermore, a [[ring of formal power series]] in one variable over a field is a PID since every ideal is of the form <math>(x^k)</math>,<!-- Probably true for any noetherian local ring with a principal maximal ideal.-->
* <math>\mathbb{Z}[i]</math>: the ring of [[Gaussian integers]],<ref>See Fraleigh & Katz (1967), p. 385, Theorem 7.8 and p. 377, Theorem 7.4.</ref>
* <math>\mathbb{Z}[\omega]</math> (where <math>\omega</math> is a primitive [[cube root]] of 1): the [[Eisenstein integers]],
* Any [[discrete valuation ring]], for instance the ring of [[p-adic integer|{{mvar|p}}-adic integers]] <math>\mathbb{Z}_p</math>.

=== Non-examples ===
Examples of integral domains that are not PIDs:

* <math>\mathbb{Z}[\sqrt{-3}]</math> is an example of a ring that is not a [[unique factorization domain]], since <math>4 = 2\cdot 2 = (1+\sqrt{-3})(1-\sqrt{-3}).</math> Hence it is not a principal ideal domain because principal ideal domains are unique factorization domains. Also, <math>\langle 2, 1+\sqrt{-3} \rangle</math> is an ideal that cannot be generated by a single element.
* <math>\mathbb{Z}[x]</math>: the ring of all polynomials with integer coefficients. It is not principal because <math>\langle 2, x \rangle</math> is  an ideal that cannot be generated by a single polynomial.
* <math>K[x, y, \ldots],</math>  the [[Polynomial_ring#Definition (multivariate case)|ring of polynomials in at least two variables]] over a ring {{mvar|K}} is not principal, since the ideal <math>\langle x, y \rangle</math> is not principal.
* Most [[ring of algebraic integers|rings of algebraic integers]] are not principal ideal domains. This is one of the main motivations behind Dedekind's definition of [[Dedekind domain]]s, which allows replacing [[unique factorization]] of elements with unique factorization of ideals. In particular, many <math>\mathbb{Z}[\zeta_p],</math> for the [[Root of unity|primitive p-th root of unity]] <math>\zeta_p,</math> are not principal ideal domains.<ref>{{Cite web|first = James|last=Milne|authorlink = James Milne (mathematician)|title=Algebraic Number Theory|url=https://www.jmilne.org/math/CourseNotes/ANT301.pdf|pages=5}}</ref> The [[Class number (number theory)|class number]] of a ring of algebraic integers gives a measure of "how far away" the ring is from being a principal ideal domain.

==Modules==
{{main|Structure theorem for finitely generated modules over a principal ideal domain}}
The key result is the structure theorem:  If ''R'' is a principal ideal domain, and ''M'' is a finitely
generated ''R''-module, then <math>M</math> is a direct sum of cyclic modules, i.e., modules with one generator.  The cyclic modules are isomorphic to <math>R/xR</math> for some <math>x\in R</math><ref>See also Ribenboim (2001), [https://books.google.com/books?id=u5443xdaNZcC&pg=PA113 p. 113], proof of lemma 2.</ref> (notice that <math>x</math> may be equal to <math>0</math>, in which case <math>R/xR</math> is <math>R</math>).

If ''M'' is a [[free module]] over a principal ideal domain ''R'', then every submodule of ''M'' is again free.<ref>[https://people.math.sc.edu/mcnulty/algebra/grad/pidfree.pdf Lecture 1. Submodules of Free Modules over a PID] math.sc.edu Retrieved 31 March 2023</ref><!-- need a better reference; for example, from a textbook --> This does not hold for modules over arbitrary rings, as the example <math>(2,X)  \subseteq \mathbb{Z}[X]</math>   of  modules over <math>\mathbb{Z}[X]</math> shows.

==Properties==
In a principal ideal domain, any two elements {{math|''a'',''b''}} have a [[greatest common divisor]], which may be obtained as a generator of the ideal {{math|(''a'', ''b'')}}.

All [[Euclidean domain]]s are principal ideal domains, but the converse is not true.
An example of a principal ideal domain that is not a Euclidean domain is the ring <math>\mathbb{Z}\left[\frac{1+\sqrt{-19}} 2\right]</math>,<ref>Wilson, Jack C. "A Principal Ring that is Not a Euclidean Ring." [[Math. Mag]] '''46''' (Jan 1973) 34-38 [https://www.jstor.org/stable/2688577]</ref><ref>George Bergman, ''A principal ideal domain that is not Euclidean - developed as a series of exercises'' [http://math.berkeley.edu/~gbergman/grad.hndts/nonEucPID.ps PostScript file]</ref> this was proved by [[Theodore Motzkin]] and was the first case known.<ref>{{Cite journal |last=Motzkin |first=Th |date=December 1949 |title=The Euclidean algorithm |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-12/The-Euclidean-algorithm/bams/1183514381.full |journal=[[Bulletin of the American Mathematical Society]] |volume=55 |issue=12 |pages=1142–1146 |doi=10.1090/S0002-9904-1949-09344-8 |issn=0002-9904|doi-access=free }}</ref> In this domain no {{mvar|q}} and {{mvar|r}} exist, with {{math|0 ≤ {{!}}''r''{{!}} < 4}}, so that <math>(1+\sqrt{-19})=(4)q+r</math>, despite <math>1+\sqrt{-19}</math> and <math>4</math> having a greatest common divisor of {{math|2}}.

Every principal ideal domain is a [[unique factorization domain]] (UFD).<ref>Proof: every prime ideal is generated by one element, which is necessarily prime. Now refer to the fact that an integral domain is a UFD if and only if its prime ideals contain prime elements.</ref><ref>Jacobson (2009), p. 148, Theorem 2.23.</ref><ref>Fraleigh & Katz (1967), p. 368, Theorem 7.2</ref><ref>Hazewinkel, Gubareni & Kirichenko (2004), [https://books.google.com/books?id=AibpdVNkFDYC&pg=PA166 p.166], Theorem 7.2.1.</ref> The converse does not hold since for any UFD {{math|''K''}}, the ring {{math|''K''[''X'', ''Y'']}}  of polynomials in 2 variables is a UFD but is not a PID. (To prove this look at the ideal generated by <math>\left\langle X,Y \right\rangle.</math> It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element.)

#Every principal ideal domain is [[noetherian ring|Noetherian]].
#In all unital rings, [[maximal ideal]]s are [[prime ideal|prime]]. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
#All principal ideal domains are [[integrally closed domain|integrally closed]].

The previous three statements give the definition of a [[Dedekind domain]], and hence every principal ideal domain is a Dedekind domain.

Let ''A'' be an integral domain, the following are equivalent.

# ''A'' is a PID.
# Every prime ideal of ''A'' is principal.<ref>{{Cite web|url=http://math.berkeley.edu/~mreyes/oka1.pdf|archive-url=https://web.archive.org/web/20100726160025/http://math.berkeley.edu/~mreyes/oka1.pdf|url-status=dead|title=T. Y. Lam and Manuel L. Reyes, A Prime Ideal Principle in Commutative Algebra|archive-date=26 July 2010|access-date=31 March 2023}}</ref>
# ''A'' is a Dedekind domain that is a UFD.
# Every finitely generated ideal of ''A'' is principal (i.e., ''A'' is a [[Bézout domain]]) and ''A'' satisfies the [[ascending chain condition on principal ideals]].
# ''A'' admits a [[Dedekind–Hasse norm]].<ref>Hazewinkel, Gubareni & Kirichenko (2004), [https://books.google.com/books?id=AibpdVNkFDYC&pg=PA170 p.170], Proposition 7.3.3.</ref>

Any [[Euclidean function|Euclidean norm]] is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID. (4) compares to:
* An integral domain is a UFD if and only if it is a [[GCD domain]] (i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals.
An integral domain is a [[Bézout domain]] if and only if any two elements in it have a gcd ''that is a linear combination of the two.'' A Bézout domain is thus a GCD domain, and (4) gives yet another proof that a PID is a UFD.

== See also ==
*[[Bézout's identity]]

== Notes ==
{{Reflist}}

==References==
* [[Michiel Hazewinkel]], Nadiya Gubareni, V. V. Kirichenko. ''Algebras, rings and modules''. [[Kluwer Academic Publishers]], 2004. {{isbn|1-4020-2690-0}}
* John B. Fraleigh, Victor J. Katz. ''A first course in abstract algebra''. Addison-Wesley Publishing Company. 5 ed., 1967. {{isbn|0-201-53467-3}}
* [[Nathan Jacobson]]. Basic Algebra I. Dover, 2009. {{isbn|978-0-486-47189-1}}
* Paulo Ribenboim. ''Classical theory of algebraic numbers''. Springer, 2001. {{isbn|0-387-95070-2}}

== External links ==
* [http://mathworld.wolfram.com/PrincipalRing.html Principal ring] on [[MathWorld]]

{{DEFAULTSORT:Principal Ideal Domain}}
[[Category:Commutative algebra]]
[[Category:Ring theory]]