Ascending chain condition on principal ideals
In
The counterpart
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 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
An integral domain A satisfies (ACCP) if and only if the polynomial ring A[t] does.[2] The analogous fact is false if A is not an integral domain.[3]
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.[4]
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
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 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
References
- ^ Nagata 1975, Lemma 2.1.
- MR 0860048.
- ^ Heinzer & Lantz 1994.
- ^ 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.
- ^ Lam 1999, pp. 230–231.
- Bass, Hyman (1960), "Finitistic dimension and a homological generalization of semi-primary rings", Trans. Amer. Math. Soc., 95 (3): 466–488, MR 0157984
- Grams, Anne (1974), "Atomic rings and the ascending chain condition for principal ideals", Proc. Cambridge Philos. Soc., 75 (3): 321–329, S2CID 120558655
- Heinzer, William J.; Lantz, David C. (1994), "ACCP in polynomial rings: a counterexample", Proc. Amer. Math. Soc., 121 (3): 975–977, MR 1232140
- Jonah, David (1970), "Rings with the minimum condition for principal right ideals have the maximum condition for principal left ideals", Math. Z., 113 (2): 106–112, S2CID 121739821
- Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, vol. 189, Berlin, New York: MR 1653294
- Nagata, Masayoshi (1975), "Some types of simple ring extensions" (PDF), Houston J. Math., 1 (1): 131–136, MR 0382248