Commutative rings
It is well known that a nonzero nonunit in a Noetherian integral domain factors into 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. But the converse is false, as shown in .) 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 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 ''S''−1''A'' is a UFD, so is ''A''. (Note that the converse of this is trivial.) An integral domain ''A'' satisfies (ACCP) if and only if the polynomial ring ''A'' 't''does. The analogous fact is false if ''A'' is not an integral domain. 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.Proof: In a Bézout domain the ACCP is equivalent to the ACC on 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. 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 : 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 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 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 between "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 orthogonal idempotents, and that ''R'' is a Dedekind finite ring.Footnotes
References
* * * * * *{{citation , last=Nagata, first= Masayoshi , title=Some types of simple ring extensions , journal=Houston Journal of Mathematics , volume=1 , year=1975 , number=1 , pages=131–136 , issn=0362-1588 , url=http://whjm.math.uzh.ch/v001n1/0131NAGATA.pdf , mr=0382248 Ideals (ring theory)