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# Properties of U-factorizations

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## Abstract

Let R be a commutative ring with identity. In this paper we examine a type of factorization called a U-factorization. We classify all possible rearrangements of U-factorizations and extend several results concerning finite factorization properties to U-factorizations. We explore the close relationship between a U-factorization in a ring R and a factorization in a related monoid, (R/∼,·), where ∼ is the associate relationship. We then examine U-factorizations in idealizations.
... With the theory of UFDs fairly extensively studied and well understood, many authors (e.g., Fletcher [19,20], Bouvier [18], Galovich [22], Anderson et al. [1,8,12], and Axtell et al. [15]) have more recently considered how best to define a "unique factorization ring" with zero divisors. The presence of zero divisors introduces several choices that must be made in formulating the definition of "unique factorization ring." ...
... Since every factorization can be rearranged (in a not necessarily unique way [2, p. 9]) to form a U-factorization [2, Proposition 4.1], (m-)atomic (resp., very (strongly) atomic, (w)p-atomic) implies (m-)U-atomic [12, p. 458] (resp., (very) strongly U-atomic, (w)p-U-atomic). However, it is a longstanding open problem whether atomic implies U-atomic [12,14,15]. When dealing with U-factorizations, we only really care about the essential parts. ...
... The main theorem of the paper will be Theorem 3.5, where we prove several nontrivial new characterizations of UWHRs. Between this theorem and others that follow from it, we will solve the previously open problems of providing complete structural descriptions of the "U-unique factorization rings" introduced by Axtell et al. [15] and the "weak (strongly) (µ-)reduced unique factorization rings" defined by Anderson, Chun, and Valdes-Leon [8]. ...
Article
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In recent decades, mathematicians have extended the definition of a unique factorization domain to rings with zero divisors in many different ways, by mixing and matching different notions of "irreducible" elements, "equivalent" factorizations, and "redundant factorizations" from the literature and deciding which elements are required to have "unique factorizations." Their study raises many natural questions such as how many distinct kinds of "unique factorization rings" one can obtain through this approach, which combinations of choices lead to equivalent notions, and if one can find nice structural characterizations of each of these notions. This paper will be answering these questions.
... In this way, we can place our later study of (U-)factorization of ideals in the appropriate context. Almost all of the definitions presented in this section are by analogy with those in [1, 13,17,18]. These papers formulate their work in ring-theoretic terms, but most of the definitions and results carry over mutatis mutandis to a more general monoid setting. ...
... Fletcher [25] focused his attention on unique U-atomic factorizations. Non-unique U-atomic factorizations were later considered in more depth by Aḡargün, Anderson, and Valdes-Leon [1]; Axtell [17]; Axtell, Forman, Roersma, and Stickles [18]; and Mooney [36]. In analogy with their (ring-theoretic) definitions, we call H U-atomic if every nonzero nonunit has a U-atomic factorization, a U-BFM if L U (x) < ∞ for each nonzero nonunit x, a U-idf-monoid if each nonzero nonunit has only finitely many irreducible essential divisors up to associates, a U-FFM if every nonzero nonunit has only finitely many U-factorizations up to U-isomorphism, a U-HFM if it is U-atomic and any two U-atomic factorizations of the same nonzero nonunit have the same essential length, and a U-UFM if each nonzero nonunit has a unique U-atomic factorization up to U-isomorphism. ...
... Again, we must emphasize that some care must be taken when reading the existing literature, due to the fact that sometimes "U-UFR" is used as we use it and sometimes it is used to mean Fletcher UFR, etc. (There are a couple minor errors caused by this inconsistency in otherwise excellent resources. For example, our definition of U-UFR is the same as in [18], and the assertion that UFR ⇒ U-UFR [18] is only true with this interpretation, but [18, Theorem 2.1] and [18,Theorem 3.3] are only true if "U-UFR" is interpreted as "Fletcher UFR".) ...
Article
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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.
... More recently, the method of U-factorization originated by C.R. Fletcher has been revisited and extended out of unique factorization rings, to a more general class of rings by M. Axtell, S. Forman, N. Roersma, and J. Stickles in [13, 14]. The purpose of this dissertation has been to study how one can effectively use the τ -factorization techniques in commutative rings with zero-divisors. ...
... Fletcher in [24, 25]. This method of factorization has been studied extensively by A.G. AˇgargünAˇgargün, D.D. Anderson, M. Axtell, N. S. Forman, N. Roersma , J. Stickles, S. Valdez-Leon in [13, 14, 1] and others. In this chapter, we extend τ -factorization to rings with zero-divisors using the method of U-factorization and in so doing synthesize this work done into a single study of what we will call τ -U-factorization. ...
... As in [14], we define U-factorization as follows. Let a ∈ R be a non-unit. ...
Thesis
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The study of factorization in integral domains has a long history. Unique factorization domains, like the integers, have been studied extensively for many years. More recently, mathematicians have turned their attention to generalizations of this such as Dedekind domains or other domains which have weaker factorization properties. Many authors have sought to generalize the notion of factorization in domains. One particular method which has encapsulated many of the generalizations into a single study is that of $\tau$-factorization, studied extensively by A. Frazier and D.D. Anderson. Another generalization comes in the form of studying factorization in rings with zero-divisors. Factorization gets quite complicated when zero-divisors are present due to the existence of several types of associate relations as well as several choices about what to consider the irreducible elements. In this thesis, we investigate several methods for extending the theory of $\tau$-factorization into rings with zero-divisors. We investigate several methods including: 1) the approach used by A.G. A\v{g}arg\"{u}n and D.D. Anderson, S. Chun and S. Valdes-Leon in several papers; 2) the method of U-factorization developed by C.R. Fletcher and extended by M. Axtell, J. Stickles, and N. Baeth and 3) the method of regular factorizations and 4) the method of complete factorizations. This thesis synthesizes the work done in the theory of generalized factorization and factorization in rings with zero-divisors. Along the way, we encounter several nice applications of the factorization theory. Using $\tau_z$-factorizations, we discover a nice relationship with zero-divisor graphs studied by I. Beck as well as D.D. Anderson, D.F. Anderson, A. Frazier, A. Lauve, and P. Livingston. Using $\tau$-U-factorization, we are able to answer many questions that arise when discussing direct products of rings. There are several benefits to the regular factorization factorization approach due to the various notions of associate and irreducible coinciding on regular elements greatly simplifying many of the finite factorization property relationships. Complete factorization is a very natural and effective approach taken to studying factorization in rings with zero-divisors. There are several nice results stemming from extending $\tau$-factorization in this way. Lastly, an appendix is provided in which several examples of rings satisfying the various finite factorization properties studied throughout the thesis are given.
... Now we investigate the notion of a U -factorization. It was introduced by Fletcher [35,36] and developed by Axtell et al. in [11] and [12]. Let S be a ring and consider a nonunit a ∈ S. By a factorization of a we mean a = a 1 · · · a s where each a i is a nonunit. ...
... Based on the proof of [11,Theorem 4.2], it is asserted in [12,Theorem 3.6] that, if R ⋉ 1 M 1 is a U -FFR, then for every nonzero nonunit d ∈ R, there are only finitely many distinct principal ideals (d, m) in R ⋉ 1 M 1 . However, a careful reading of this proof shows that the case of ideals (d, m) with dM 1 = 0 should be also treated. ...
Article
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The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra. In this paper we extend this classical ring construction by associating a ring to a ring $R$ and a family $M=(M_i)_{i=1}^{n}$ of $n$ $R$-modules for a given integer $n\geq 1$. We call this new ring construction an $n$-trivial extension of $R$ by $M$. In particular, the classical trivial extension will be just the $1$-trivial extension. Thus we generalize several known results on the classical trivial extension to the setting of $n$-trivial extensions and we give some new ones. Various ring-theoretic constructions and properties of $n$-trivial extensions are studied and a detailed investigation of the graded aspect of $n$-trivial extensions is also given. We end the paper with an investigation of various divisibily properties of $n$-trivial extensions. In this context several open questions arise.
... Now we investigate the notion of a U -factorization. It was introduced by Fletcher [35,36] and developed by Axtell et al. in [11] and [12]. Let S be a ring and consider a nonunit a ∈ S. By a factorization of a we mean a = a 1 · · · a s where each a i is a nonunit. ...
... Based on the proof of [11,Theorem 4.2], it is asserted in [12,Theorem 3.6] that, if R ⋉ 1 M 1 is a U -FFR, then for every nonzero nonunit d ∈ R, there are only finitely many distinct principal ideals (d, m) in R ⋉ 1 M 1 . However, a careful reading of this proof shows that the case of ideals (d, m) with dM 1 = 0 should be also treated. ...
Research
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Rocky Mountain J. Math. Volume 47, Number 8 (2017), 2439-2511. https://projecteuclid.org/euclid.rmjm/1517648596
... For a detailed study of these factorization properties see [2,3]. In [7], Axtell and et al. study U-hal ff actorial rings (U-HFR) without the assumption of atomicity. In Section 2, we briefly review this factorization property. ...
... By a factorization of a we mean a = a 1 a 2 ... a n where each a i is a nonunit. Another type of factorization is a U-factorization which was introduced in [1] and extensively examined in [6,7,12]. Hence every U-decomposition of a nonunit x ∈ R is a U-factorization of x. ...
... Corollary 3. 13. Let R be a ring, {X λ } λ∈Λ be a nonempty family of algebraically independent indeterminates, α ∈ {UFD, UFR, v-UHR, UHR}, and β ∈ {UWHR, factorial ring, pfactorial ring}. ...
Article
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Several different versions of "factoriality" have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of "unique factorization rings." Our work generalizes Anderson et al.'s results about "unique factorization" in R[X], Gilmer and Parker's characterization of factorial monoid domains, and Hardy and Shores's classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is "restricted cancellative" or satisfies various "(restricted) ideal cancellation laws."
... Any commutative ring R is called a bounded factorization ring (BFR) if for each nonzero nonunit a R ∈ , there exist a natural number ( ) N a so that for any factorization It is clear that UFR ⇒ U-HFR. For these factorization properties see [1,7,8]. ...
Article
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We investigate the factorization properties on the polynomial extension [ ] A X of A where A is a UFR and show that [ ] A X is a U-BFR for any UFR A . We also consider the ring structure [ ] A XI X + where A is a UFR. A bir TÇA halka (tektürlü çarpanlara ayrılabilen halka) olmak üzere A 'nın polinom genişlemesi [ ] A X üzerindeki çarpanlara ayırma özelliklerini araştırıyoruz ve herhangi bir TÇA halka A için [ ] A X 'in U-KÇA halka (U-Kısıtlı Çarpanlarına Ayrılabilen Halka) olduğunu gösteriyoruz. A bir TÇA olmak üzere [ ] A XI X + yapısındaki halkaları da göz önüne alıyoruz. Anahtar Sözcükler : Çarpanlara ayırma, Polinom halkaları.
Preprint
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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.]
Article
Full-text available
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.”
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