Editing Principal ideal domain (section)

==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.