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In [[abstract algebra]], the [[ascending chain condition]] can be applied to the [[poset]]s of principal left, principal right, or principal two-sided ideals of a [[ring (mathematics)|ring]], partially ordered by [[inclusion (set theory)|inclusion]]. The ascending '''ascending chain condition on principal ideals''' (abbreviated to '''ACCP''') is satisfied if there is no infinite strictly ascending chain of [[principal ideal]]s of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.
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The counterpart [[descending chain condition]] may also be applied to these posets, however there is currently no need for the terminology "DCCP" since such rings are already called left or right [[perfect ring]]s. (See Noncommutative ring section below.)
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[[Noetherian ring]]s (e.g. [[principal ideal domain]]s) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notably [[unique factorization domain]]s and left or right perfect rings.
 
==Commutative rings==
It is well known that a nonzero nonunit in a Noetherian integral domain factors into [[irreducible element|irreducibles]]. The proof of this relies on only (ACCP) not (ACC), so in any integral domain with (ACCP), an irreducible factorization exists. (In other words, any integral domains with (ACCP) are [[atomic domain|atomic]]. But the converse is false, as shown in {{harv|Grams|1974}}.) Such a factorization may not be unique; the usual way to establish uniqueness of factorizations uses [[Euclid's lemma]], which requires factors to be [[prime element|prime]] rather than just irreducible. Indeed one has the following characterization: let ''A'' be an integral domain. Then the following are equivalent.
# ''A'' is a UFD.
# ''A'' satisfies (ACCP) and every irreducible of ''A'' is prime.
# ''A'' is a [[GCD domain]] satisfying (ACCP).
 
The so-called '''Nagata criterion''' holds for an integral domain ''A'' satisfying (ACCP): Let ''S'' be a [[multiplicatively closed subset]] of ''A'' generated by prime elements. If the [[localization of a ring|localization]] ''S''<sup>&minus;1</sup>''A'' is a UFD, so is ''A''. {{harv|Nagata| 1975, Lemma 2.1}} (Note that the converse of this is trivial.)
 
An integral domain ''A'' satisfies (ACCP) if and only if the polynomial ring ''A''[''t''] does.{{Citation needed|date=April 2009}} The analogous fact is false if ''A'' is not an integral domain. {{harv| Heinzer, Lantz|1994}}
 
An [[integral domain]] where every finitely generated ideal is principal (that is, a [[Bézout domain]]) satisfies (ACCP) if and only if it is a [[principal ideal domain]].<ref>Proof: In a Bézout domain the ACCP is equivalent to the ACC on [[finitely generated module|finitely generated ideals]], but this is known to be equivalent to the ACC on ''all'' ideals. Thus the domain is Noetherian and Bézout, hence a principal ideal domain.</ref>
 
The ring '''Z'''+''X'''''Q'''[''X''] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the chain of principal ideals
:<math>(X) \subset (X/2) \subset (X/4) \subset (X/8), ...</math>
is non-terminating.
 
==Noncommutative rings==
In the noncommutative case, it becomes necessary to distinguish the '''right ACCP''' from '''left ACCP'''.  The former only requires the poset of ideals of the form ''xR'' to satisfy the ascending chain condition, and the latter only examines the poset of ideals of the form ''Rx''.
 
A theorem of [[Hyman Bass]] in {{harv|Bass|1960}} now known as "Bass' Theorem P" showed that the ''descending chain condition'' on principal ''left'' ideals of a ring ''R'' is equivalent to ''R'' being a ''right'' [[perfect ring]]. D. Jonah showed in {{harv|Jonah|1970}} that there is a side-switching connection between the ACCP and perfect rings. It was shown that if ''R'' is right perfect (satisfies right DCCP), then ''R'' satisfies the left ACCP, and symmetrically, if ''R'' is left perfect (satisfies left DCCP), then it satisfies the right ACCP. The converses are not true, and the above switches from "left" and "right" are not typos.
 
Whether the ACCP holds on the right or left side of ''R'', it implies that ''R'' has no infinite set of nonzero [[idempotence#Idempotent ring elements|orthogonal idempotent]]s, and that ''R'' is a [[Dedekind finite ring]]. {{harv|Lam|1999, p.230-231}}
 
== References ==
{{reflist}}
*{{citation  |author=Bass, Hyman  |title=Finitistic dimension and a homological generalization of semi-primary rings  |journal=Trans. Amer. Math. Soc.  |volume=95  |year=1960  |pages=466–488  |issn=0002-9947  |mr=0157984 }}
*{{Citation  |author=Grams, Anne  |title=Atomic rings and the ascending chain condition for principal ideals  |journal=Proc. Cambridge Philos. Soc.  |volume=75  |year=1974  |pages=321–329  |mr=0340249 }}
*{{citation  |author1=Heinzer, William J.  |author2=Lantz, David C.  |title=ACCP in polynomial rings: a counterexample  |journal=Proc. Amer. Math. Soc.  |volume=121  |year=1994  |number=3
  |pages=975–977  |issn=0002-9939  |mr=1653294 |jstor=2160301  |doi=10.2307/2160301 }}
*{{citation  |author=Jonah, David  |title=Rings with the minimum condition for principal right ideals have the maximum condition for principal left ideals  |journal=Math. Z.  |volume=113  |year=1970  |pages=106–112  |issn=0025-5874  |mr=0260779 }}
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}}
*{{citation  |author=Nagata, Masayoshi  |title=Some types of simple ring extensions  |journal=Houston J. Math.  |volume=1  |year=1975  |number=1  |pages=131–136  |issn=0362-1588
  |url=http://hjm.math.unizh.ch/v001n1/0131NAGATA.pdf  |mr=0382248 }}
 
[[Category:Ring theory]]

Latest revision as of 17:53, 19 July 2014

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