Unique factorization domains (a.k.a. UFD's or factorial domains) have a central place in mathematics. Some of the first theorems a student of abstract algebra encounters are that certain familiar rings are UFD's. The Fundamental Theorem of Arithmetic states that the ring of integers ℤ is a UFD. The well-known implications Euclidean ⇒ PID ⇒ UFD yield that for a field K, K[X] is a UFD. Ideal theory owes its start to the fact that while a ring of algebraic integers need not be a UFD, unique factorization can be restored via factorization into prime ideals. At a more advanced level, one of the first triumphs of homological algebra was that a regular local ring is a UFD [29]. Perhaps the first name that comes to mind when thinking about UFD's is Pierre Samuel. He has done fundamental work on Euclidean rings [76], UFD's, and power series rings over UFD's [72]. The reader is referred to his survey article [75] and monographs [73], [74]. His collected works [77], which contain a number of papers related to unique factorization, have recently been published in the Queen's Papers in Pure and Applied Mathematics series. Moreover, several of his students have done work dealing with unique factorization and its extensions to commutative rings with zero divisors and to modules.

Objective: Radiographic joint space width (JSW) has been a standard for measuring knee osteoarthritis (OA) structural change. Limitations in the responsiveness of this approach might be overcome by instead measuring 3D JSW on weight-bearing CT (WBCT). This study compared the responsiveness of 3D JSW measurements using WBCT with the responsiveness of radiographic 2D JSW.Design: Standing, fixed -flexion knee radiographs (XR) and WBCT were acquired ancillary to the 144-and 168-month Multicenter Osteoarthritis Study visits. Tibiofemoral JSW was measured on both XR and WBCT. Responsiveness to change was defined by the standardized response mean (SRM) for change in JSW (1) at predetermined mediolateral locations (JSWx) on both modalities and (2) in the following subregions measured on WBCT images: central medial and lateral femur (CMF/CLF) and tibia (CMT/CLT), and anterior and posterior tibia (AMT/ALT, PMT/MLT).Results: Baseline and 24-month follow-up JSWx measurements were completed for 265 participants (58.1% women). Responsiveness of 3D JSWx for medial tibiofemoral compartment on coronal WBCT (SRM range:-0.18,-0.24) exceeded that for 2D JSWx (-0.10,-0.16). Responsiveness of 3D JSW sub -regional mean (-0.06,-0.36) and maximal (-1.14,-1.75) CMF and CMT and maximal CLF/CLT 3D JSW changes were statistically significantly greater in comparison with respective medial and lateral 2D JSWx (P < 0.002).Conclusions: Subregional 3D JSW on WBCT is substantially more responsive to 24-month changes in tibiofemoral joint structure compared to radiographic measurements. Use of subregional 3D JSW on WBCT could enable improved detection of OA structural progression over a 24-month duration in comparison with measurements made on XR.(c) 2022 Osteoarthritis Research Society International. Published by Elsevier Ltd. All rights reserved.

Let D be a commutative integral domain without identity. We show that every principal ideal of D is a product of prime ideals if and only if D[I] is a quasilocal UFD with D as its unique maximal ideal.

We define a subclass of the class of periodic rings by calling a ring R an (n, m)-ring if for each $$x\in R$$ there exist two fixed positive integers n, m with $$n>m$$ such that $$x^n=x^m$$ . We study when an (n, m)-ring is either Boolean, potent, reduced or commutative.

An extension [Formula: see text] of commutative rings is strongly inert (respectively, inert) if for nonzero [Formula: see text], [Formula: see text] implies [Formula: see text] (respectively, there exists a unit [Formula: see text] with [Formula: see text]). We investigate strongly inert and inert extensions and various related extensions, where we either require the product [Formula: see text] to be nonzero or that [Formula: see text] and [Formula: see text] be nonzerodivisors. Special emphasis is given to the relationship between factorization properties of [Formula: see text] and [Formula: see text].

Purpose: Intra-articular fractures (IAFs) trigger elevated synovial fluid concentrations of pro-inflammatory cytokines and degradative enzymes. Some individuals appear to exhibit a phenotypic dysregulated inflammatory response after joint injury characterized by increased pro-inflammatory markers in the absence of an innate anti-inflammatory response. A dysregulated inflammatory response may increase the risk for early joint degeneration associated with post-traumatic osteoarthritis (PTOA). Prior work showed that IAF severity influences PTOA risk, and this important variable has not been accounted for in inflammatory phenotyping studies.

In this paper, we introduce ∗-almost independent rings of Krull type (∗-almost IRKTs) and ∗-almost generalized Krull domains (∗-almost GKDs) in the general theory of almost factoriality, neither of which need be integrally closed. This fills a gap left in [D. D. Anderson and M. Zafrullah, On∗-Semi-Homogeneous Integral Domains, Advances in Commutative Algebra (Springer, Singapore, 2019)]. We characterize them by ∗-almost super-SH domains, where a domain [Formula: see text] is called a ∗-almost super-SH domain if every nonzero proper principal ideal of [Formula: see text] is a ∗-product of ∗-almost super-homogeneous ideals. We prove that (1) a domain [Formula: see text] is a ∗-almost IRKT if and only if [Formula: see text] is a ∗-almost super-SH domain, (2) a domain is a ∗-almost GKD if and only if [Formula: see text] is a type 1 ∗-almost super-SH domain and (3) a domain [Formula: see text] is a ∗-almost IRKT and an AGCD-domain if and only if [Formula: see text] is a ∗-afg-SH domain. Further, we characterize them by their integral closures. For example, we prove that a domain [Formula: see text] is an almost IRKT if and only if [Formula: see text] is a root extension with [Formula: see text] [Formula: see text]-linked under [Formula: see text] and [Formula: see text] is an IRKT. Examples are given to illustrate the new concepts.

Let [Formula: see text] be a commutative ring. A polynomial [Formula: see text] is an annihilating content (AC) polynomial if [Formula: see text] where [Formula: see text] and [Formula: see text] is a nonzerodivisor and [Formula: see text] is an EM-ring if each [Formula: see text] is an AC polynomial. In this paper, we investigate AC polynomials and EM-rings and extensions to modules. For example, we show that [Formula: see text] is an EM-ring if and only if every linear polynomial is an AC polynomial if and only if for each finitely generated ideal [Formula: see text] of [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is a finitely generated ideal of [Formula: see text] with [Formula: see text]. Special attention is given to the case when [Formula: see text] is Noetherian where such rings are characterized by the property that each associated prime is principal.

Purpose: Kellgren-Lawrence (KL) grading remains the clinical standard for assessing tibiofemoral OA severity from weight bearing radiographs, evaluating features such as joint space narrowing. The qualitative nature of KL grading introduces inter- and intra-observer variability, and the 2D projection of the 3D joint surface seen on plain radiographs limits reliability for assessing joint space narrowing. An objective and fully automated method to quantify joint space narrowing would reduce variability in OA severity grading.

Radiography is the current standard for assessing osseous structural knee OA, but it is a 2D image of a 3D structure and superimposition of bone impairs ability to sensitively detect changes. Weight-bearing CT (WBCT) allows improved visualization of OA features, shows greater correlation with central medial cartilage damage on MRI and has demonstrated excellent test-retest reliability, suggesting that WBCT may address shortcomings of radiographs. To determine the concurrent validity of change in 3D JSW on WBCT vs. change in 2D JSW on radiographs with radiographic progression defined by semiquantitative grading using of cartilage damage worsening on MRI (MOAKS) over 24 months. Radiographs, WBCT and MRI were obtained from 259 participants from the Multicenter Osteoarthritis Study (MOST) at baseline and 24-month follow-up (including one knee per participant). Radiographs were evaluated for change in 2D JSW at predetermined locations in the medial and lateral compartments. Change in 3D JSW on WBCT assessed JSW in each subregion (Figure). MRI scans were assessed using MRI Osteoarthritis of the Knee Score for cartilage morphology (MOAKS-CM). Concurrent validity was assessed by associations of JSW change on radiographs and WBCT with structural worsening of MOAKS-CM using logistic regression. Area under the receiver operating characteristic (AUROC) curve was calculated to compare the concurrent validity of 2D JSW vs. 3D JSW by compartment and subregion. A "no exposure" model was reported using only demographic variables. Exposure variables in each model contained either 2D JSWx or 3D JSW. Two separate parameter types were utilized for 3D JSW exposure, maximum change values and change in mean values. In addition to the main exposure JSW parameter, each model adjusted for demographic covariates (age, sex and BMI). JSWx=.225 was used when comparing to medial 3D JSW subregional change and JSWx=.750 was used when comparing to lateral 3D JSW subregional change. Mean age was 63.2 ± 9.0 years, BMI was 28.2±4.9 kg/m2, 57% were women and knees had KL Grade 2-3 at baseline for 60/259. For change in medial compartment MOAKS-CM, 3D Mean (AUROC=0.681) and Maximum (AUROC=0.682) were not superior to 2D JSW (AUROC=0.713). For change in lateral compartment MOAKS-CM, 3D Mean (AUROC=0.757) and Maximum (AUROC=0.730) showed no statistical difference with 2D JSW (AUROC=0.695). In subregional analyses, there was no significant difference between the association of change in MOAKS-CM and change in either 3D JSW (Mean, Maximum) or 2D JSW. In this study of the association of knee OA structural progression assessed by cartilage morphology on MRI, change in 3D JSW parameters measured on WBCT and 2D JSW on radiographs, WBCT did not demonstrate superior concurrent validity with worsening MOAKS-CM compared to change in 2D JSW assessed on knee radiographs. Thus, diagnostic value of this implementation of 3D JSW remained similar to that for 2D JSW. National Institutes of Health, University of Kansas (R01AR071648), University of Iowa (U01AG18832) and University of California-San Francisco (U01AG19069). AG and FWR are shareholders of BICL, LLC. AG is consultant to Pfizer, MerckSerono, TissueGene, Novartis, Regeneron and AstraZeneca. FWR is consultant to Grünenthal. NS is a consultant for Integra BioLife, Trice Medical and Pacira Biosciences. Others authors have no conflicts of interest to disclose. The authors would like to thank participants and staff of the MOST study. CORRESPONDENCE ADDRESS: [email protected]

Let $D$ be an integral domain. Then $D$ is an almost valuation (AV-)domain if for $a, b\in D\setminus \{0\}$ there exists a natural number $n$ with $a^{n}\mid b^{n}$ or $b^{n}\mid a^{n}$. AV-domains are closely related to valuation domains, for example, $D$ is an AV-domain if and only if the integral closure $\bar{D}$ is a valuation domain and $D\subseteq \bar{D}$ is a root extension. In this note we explore various generalizations of DVRs (which we might call almost DVRs) such as Noetherian AV-domains, AV-domains with $\bar{D}$ a DVR, and quasilocal and local API-domains (i.e., for $\{a_{\alpha}\}_{\alpha\in \Lambda}\subseteq D$, there exists an $n$ with $(\{a_{\alpha}^{n}\}_{\alpha\in \Lambda})$ principal). The structure of complete local AV-domains and API-domains is determined.

In this corrigendum, we give an example showing that the implication (2) double right arrow (1) of Proposition 4 is not true in general. Also, we provide the correct version of Proposition 4.

Abstract In this corrigendum, we give an example showing that the implication of Proposition 4 is not true in general. Also, we provide the correct version of Proposition 4.

We study modules whose endomorphism rings possess various forms of von Neumann regularity. We characterize these "regularity" properties for several classes of modules, including completely decomposable modules and finitely presented modules over commutative rings. We generalize many of the classic results about abelian groups with regular endomorphism rings to modules over one-dimensional commutative rings with Noetherian spectrum. To facilitate this study, we define a module M over a commutative ring R to be weakly endoregular if xM and (0:M(x)) are direct summands of M for each x∈R. We give several characterizations of weak endoregularity for various classes of modules.

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.

We show that if R is a ring such that for each x is an element of R there exist two natural numbers n(x) and m(x) of opposite parity with x(n(x)) = x(m(x)), then R is commutative. This extends the classical famous theorem of Jacobson [Ann. of Math. 46 (1945), p. 695-707] for commutativity of potent rings.

Ye defined a ring to be semiclean if every element of it can be written as a sum of a unit element and a periodic element. In this paper we generalize the notion of a semiclean ring to an almost semiclean ring. A ring R is said to be almost semiclean if each element is a sum of a regular element and a periodic element. We discuss some basic properties of almost semiclean rings. For example, R is almost semiclean if and only if the polynomial ring over R is almost semiclean. We also discuss when the idealization is almost semiclean. Finally, we give examples which distinguish almost semiclean rings from other classes of rings. Communicated by Silvana Bazzoni

Let [Formula: see text] be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of [Formula: see text]. Special attention is given to the set of atoms in [Formula: see text] and to the existence of atoms in [Formula: see text]. While our main interest is in local Cohen–Kaplansky (CK) domains (atomic integral domains with only finitely many nonassociate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight non-associate atoms having an atom in [Formula: see text].

Let D be an integral domain and star a star operation defined on D. We say that D is a star-power conductor domain (star-PCD) if for each pair a, b is an element of D\(0) and for each positive integer n we have Da(n) boolean AND Db(n) = ((Da boolean AND Db)(n))(star): We study star-PCDs and characterize them as root closed domains satisfying ((a, b)(n))(-1) = (((a, b)(-1))(n))(star) for all nonzero a, b and all natural numbers n >= 1. From this it follows easily that Prufer domains are d-PCDs (where d denotes the trivial star operation), and v-domains (e.g. Krull domains) are v-PCDs. We also consider when a star-PCD is completely integrally closed, and this leads to new characterizations of Krull domains. In particular, we show that a Noetherian domain is a Krull domain if and only if it is a w-PCD.