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Concept
Ascending chain condition on principal ideals
Summary
In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals (abbreviated to ACCP) is satisfied if there is no infinite strictly ascending chain of principal ideals of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.
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 rings. (See below.)
Noetherian rings (e.g. principal ideal domains) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notably unique factorization domains and left or right perfect 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−1A 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.
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