Ascending chain condition on principal ideals

Ascending chain condition on principal ideals

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 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 Noncommutative ring section 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.

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 (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 rather that just irreducible. Indeed one has the following characterization: let A be an integral domain. Then the following are equivalent.

  1. A is a UFD.
  2. A satisfies (ACCP) and every irreducible of A is prime.
  3. 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. (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] The analogous fact is false if A is not an integral domain. (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.[1]

The ring Z+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

(X) \subset (X/2) \subset (X/4) \subset (X/8), ...

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 (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 (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 orthogonal idempotents, and that R is a Dedekind finite ring. (Lam 1999, p.230-231)


  1. ^ 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.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Ascending chain condition — The ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings.[1][2][3] These conditions played an important role… …   Wikipedia

  • Bézout domain — In mathematics, a Bézout domain is an integral domain which is, in a certain sense, a non Noetherian analogue of a principal ideal domain. More precisely, a Bézout domain is a domain in which every finitely generated ideal is principal. A… …   Wikipedia

  • GCD domain — A GCD domain in mathematics is an integral domain R with the property that any two non zero elements have a greatest common divisor (GCD). Equivalently, any two non zero elements of R have a least common multiple (LCM). [cite book|author=Scott T …   Wikipedia

  • Noetherian ring — In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it… …   Wikipedia

  • Glossary of ring theory — Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. Contents 1 Definition of a ring 2 Types of… …   Wikipedia

  • Liste des publications d'Emmy Noether — Emmy Noether (1882 1935) est une mathématicienne allemande spécialiste de l algèbre. Cet article est une liste des publications qui ont fait sa renommée. Sommaire 1 Première époque (1908–1919) 2 Deuxième époque (1920–1926) 3 Troisiè …   Wikipédia en Français

  • Module (mathematics) — For other uses, see Module (disambiguation). In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring. Modules also… …   Wikipedia

  • Unique factorization domain — In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements, analogous to the fundamental theorem of… …   Wikipedia

  • List of commutative algebra topics — Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative… …   Wikipedia

  • Commutative ring — In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Some specific kinds of commutative rings are given with …   Wikipedia