Inabstract algebra, theascending chain condition can be applied to theposets of principal left, principal right, or principal two-sided ideals of aring, partially ordered byinclusion. Theascending chain condition on principal ideals (abbreviated toACCP) is satisfied if there is no infinite strictly ascending chain ofprincipal ideals of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.
The counterpartdescending 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 rightperfect rings. (See§ Noncommutative rings below.)
Noetherian rings (e.g.principal ideal domains) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notablyunique factorization domains and left or right perfect rings.
It is well known that a nonzero nonunit in a Noetherian integral domain factors intoirreducibles. 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) areatomic. But the converse is false, as shown in (Grams 1974).) Such a factorization may not be unique; the usual way to establish uniqueness of factorizations usesEuclid's lemma, which requires factors to beprime rather than just irreducible. Indeed, one has the following characterization: letA be an integral domain. Then the following are equivalent.
The so-calledNagata criterion holds for an integral domainA satisfying (ACCP): LetS be amultiplicatively closed subset ofA generated by prime elements. If thelocalizationS−1A is a UFD, so isA.[1] (Note that the converse of this is trivial.)
An integral domainA satisfies (ACCP) if and only if the polynomial ringA[t] does.[2] The analogous fact is false ifA is not an integral domain.[3]
Anintegral domain where every finitely generated ideal is principal (that is, aBézout domain) satisfies (ACCP) if and only if it is aprincipal ideal domain.[4]
The ringZ+XQ[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.
In the noncommutative case, it becomes necessary to distinguish theright ACCP fromleft ACCP. The former only requires the poset of ideals of the formxR to satisfy the ascending chain condition, and the latter only examines the poset of ideals of the formRx.
A theorem ofHyman Bass in (Bass 1960) now known as "Bass' Theorem P" showed that thedescending chain condition on principalleft ideals of a ringR is equivalent toR being arightperfect ring. D. Jonah showed in (Jonah 1970) that there is a side-switching connection between the ACCP and perfect rings. It was shown that ifR is right perfect (satisfies right DCCP), thenR satisfies the left ACCP, and symmetrically, ifR 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 ofR, it implies thatR has no infinite set of nonzero orthogonalidempotents, and thatR is aDedekind finite ring.[5]