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Finite factorization properties in commutative monoid rings with zero divisors
Abstract
Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about "finite factorization" properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of "finite factorization rings."
[This is the preprint of our paper with the same title that is now published in Communications in Algebra. This is the version before peer review. The published version has no changes in its results, but the presentation was changed a little bit and some more examples were added, so the numbering on the results is different in the final version. I can provide a link to get an eprint of the published version to anyone who needs it.]
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Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whether the monoid ring R[x;M] is atomic provided that both M and R are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for M=N0: he constructed an atomic integral domain R such that the polynomial ring R[x] is not atomic. However, the question of whether a monoid algebra F[x;M] over a field F is atomic provided that M is atomic has been open since then. Here we offer a negative answer to this question. First, we exhibit for any infinite cardinal κ a torsion-free atomic monoid M of rank κ satisfying that the monoid domain R[x;M] is not atomic for any integral domain R. Then for every n≥2 and for each field F of finite characteristic we find a torsion-free atomic monoid M of rank n such that F[x;M] is not atomic. Finally, we construct a torsion-free atomic monoid M of rank 1 such that Z2[x;M] is not atomic.
We study some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study here consists of all atomic monoids of the form Sr:=〈rn|n∈N0〉, where r is a positive rational. As the atomic monoids Sr are nicely generated, we are able to give detailed descriptions of many of their factorization invariants. One distinguishing characteristic of Sr is that all its sets of lengths are arithmetic sequences of the same distance, namely |a−b|, where a,b∈N are such that r=a/b and gcd(a,b)=1. We prove this, and then use it to study the elasticity and tameness of Sr.
We study the factorization of ideals of a commutative ring, defining multiple different kinds of "nonfactorable" ideals and several "factorability" properties weaker than unique factorization. We characterize (some of) these notions, determine the implications between them, and give several examples to illustrate the differences. We also examine how these properties behave with respect to localization, direct products, idealizations, polynomial rings, monoid domains, (generalized) power series rings, and the classical D + M construction. Along the way, we give some new characterizations of the finite superideal rings introduced by A.J. Hetzel and A.M. Lawson.
We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several "U-factorability" properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. For example, we show that a ring is a general ZPI-ring if and only if its monoid of ideals has unique factorization in the sense of Fletcher. We also examine how these "U-factorability" properties behave with respect to several ring-theoretic constructions.
We perform an in-depth study of several different cancellation properties for modules. Among those we consider are (half) (weak) cancellation modules, restricted cancellation modules, and (half) join principal modules. We also investigate which commutative rings have every nonzero (finitely generated) ideal (respectively, module) satisfying some cancellation property.
An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let R be an integral domain. We say that R is a bounded factorization domain if it is atomic and for every nonzero nonunit \(x \in R\), there is a positive integer N such that for any factorization \(x = a_1 \cdots a_n\) of x into irreducibles \(a_1, \dots , a_n\) in R, the inequality \(n \le N\) holds. In addition, we say that R is a finite factorization domain if it is atomic and every nonzero nonunit in R factors into irreducibles in only finitely many ways (up to order and associates). The notions of bounded and finite factorization domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in their systematic study of factorization in atomic integral domains. In this chapter, we present some of the most relevant results on bounded and finite factorization domains.KeywordsBFDBounded factorization domainFFDFinite factorization domainFactorizationD+M constructionHFDACCPAtomic domainMonoid domain2010 Mathematics Subject ClassificationPrimary: 13A0513F15Secondary: 13A1513G05
Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.
An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a bound for the number of atoms in each of its factorizations. These two properties have been systematically studied since being introduced by Anderson, Anderson, and Zafrullah in 1990. Noetherian domains satisfy the BFP, while Dedekind domains satisfy the FFP. It is well known that for commutative cancellative monoids (and, in particular, for integral domains) FFP ⇒ BFP ⇒ ACCP ⇒ atomic. For n≥2, we show that each of these four properties transfers back and forth between an information semialgebra S (certain commutative cancellative semiring) and its multiplicative monoid Tn(S)• (which consists of n×n upper triangular matrices over S). We also show that a similar transfer behavior takes place if one replaces Tn(S)• by its submonoid Un(S) consisting of upper triangular matrices with units along their main diagonals. As a consequence, we find that the atomic chain FFP ⇒ BFP ⇒ ACCP ⇒ atomic also holds for the two classes comprising the noncommutative monoids Tn(S)• and Un(S). Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible.
Let K be any field. The division algorithm plays a key role in studying the basic algebraic structure of K[X]. While the division algorithm implies that all the ideals of K[X] are principal, we show that subrings of K[X] satisfying a slightly weaker version of the division algorithm produce ideals that while not principal, are still finitely generated. Our construction leads to an example for each positive integer n of an integral domain with the n, but not the n − 1, generator property.
Dedicated to the Memory of Nick Vaughan
Every day, 34 million Chicken McNuggets are sold worldwide. At most McDonalds locations in the United States today, Chicken McNuggets are sold in packs of 4, 6, 10, 20, 40, and 50 pieces. However, shortly after their introduction in 1979 they were sold in packs of 6, 9, and 20. The use of these latter three numbers spawned the so-called Chicken McNugget problem, which asks: "what numbers of Chicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces?" In this paper, we present an accessible introduction to this problem, as well as several related questions whose motivation comes from the theory of non-unique factorization.