Ring Extensions Research Articles

On extensions of matrix rings with skew Hochschild 2-cocycles

We study structures of Hochschild 2-cocycles related to endomorphisms and introduce a skew Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible (or semicommutative) rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.

ON INTEGRALITY AND GOING-DOWN INSIDE THE FIXED RING OF A MONOID RING

An example is given of a finitely generated abelian torsion-free monoid S on which the group G with two elements acts via semigroup automorphisms such that for any field K, when the given action is extended so that G acts on the monoid ring K[X; S] via ring automorphisms that fix K elementwise, the ring extension K[X; SG] ⊆ (K[X; S])G is not integral and does not satisfy the going-down property.

A tower condition characterizing normality

We define left relative H-separable tower of rings and continue a study of these begun by Sugano. It is proven that a progenerator extension has right depth 2 if and only if the ring extension together with its right endomorphism ring is a left rel- ative H-separable tower. In particular, this applies to twisted or ordinary Frobenius extensions with surjective Frobenius homomor- phism. For example, normality for Hopf subalgebras of finite- dimensional Hopf algebras is also characterized in terms of this tower condition.

ChemInform Abstract: Geminal Bis‐ and Tristriazoles: Long‐Known but Still Uncommon Heterocyclic Entities

AbstractReview: [57 refs.

MODULES WITH FINITELY MANY SUBMODULES

We characterize ring extensions R ⊂ S having FCP (FIP), where S is the idealization of some R-module. As a by-product we exhibit characterizations of the modules that have finitely many submodules. Our tools are minimal ring morphisms, while Artinian conditions on rings are ubiquitous.

Note on Morita Equivalence in Ring Extensions

It seems that Morita invariance judges of the importance of classes of ring extensions concerned. Miyashita introduced the notion of Morita equivalence in ring extensions, and he showed that the classes of G-Galois extensions and Frobenius extensions are Morita invariant. After that, Ikehata showed that the classes of separable extensions, Hirata separable extensions, symmetric extensions, and QF-extensions are Morita invariant. In this article, we shall prove that the classes of several extensions are Morita invariant. Further, we will give an example of the class of ring extensions which is not Morita invariant.

Ramification theory for Artin–Schreier extensions of valuation rings

The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin–Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case.

Image texture and radiation dose properties in CT.

The aim of this study was to compare image noise properties of GE Discovery HD 750 and Toshiba Aquilion ONE. The uniformity section of a Catphan 600 image quality assurance phantom was scanned with both scanners, at different dose levels and with extension rings simulating patients of different sizes. 36 datasets were obtained and analyzed in terms of noise power spectrum. All the results prove that introduction of extension rings significantly altered the image quality with respect to noise properties. Without extension rings, the Toshiba scanner had lower total visible noise than GE (with GE as reference: FC18 had 82% and FC08 had 80% for 10 mGy, FC18 had 77% and FC08 74% for 15 mGy, FC18 had 80% and FC08 77% for 20 mGy). The total visible noise (TVN) for 20 and 15 mGy were similar for the phantom with the smallest additional extension ring, while Toshiba had higher TVN than GE for the 10 mGy dose level (120% FC18, 110% FC08). For the second and third ring, the GE images had lower TVN than Toshiba images for all dose levels (Toshiba TVN is greater than 155% for all cases). The results indicate that GE potentially has less image noise than Toshiba for larger patients. The Toshiba FC18 kernel had higher TVN than the Toshiba FC08 kernel with additional beam hardening correction for all dose levels and phantom sizes (120%, 107%, and 106% for FC18 compared to 110%, 98%, and 97%, for FC08, for 10, 15 and 20 mGy doses, respectively).PACS number(s): 87.57.Q‐, 87.57.nf, 87.57.C‐

Topology, intersections and flat modules

It is well known that, in general, multiplication by an ideal I I does not commute with the intersection of a family of ideals, but that this fact holds if I I is flat and the family is finite. We generalize this result by showing that finite families of ideals can be replaced by compact subspaces of a natural topological space, and that ideals can be replaced by submodules of an epimorphic extension of a base ring. As a particular case, we give a new proof of a conjecture by Glaz and Vasconcelos.

NR-Clean Rings

For an associative ring R with identity, we define an element in R to be NR-clean if it is the sum of a (von Neumann) regular and a nilpotent. R is called an NR-clean ring if every element of R is NR-clean. This class of rings lies properly between nil clean rings and UR rings. In this paper, we give many examples and properties of NR-clean rings and investigate the behavior of these properties under various ring extensions. We also define and study uniquely NR-clean rings and give a characterization of such rings. Finally, we justify some conditions under which the group rings are NR-clean.

Idempotent lifting and ring extensions

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring [Formula: see text] for which idempotents lift modulo the Jacobson radical [Formula: see text], but idempotents do not lift modulo [Formula: see text]. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if [Formula: see text] and [Formula: see text] are ideals of [Formula: see text] for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over [Formula: see text]. On the positive side, if [Formula: see text] and [Formula: see text] are enabling ideals in [Formula: see text], then [Formula: see text] is also an enabling ideal. We show that if [Formula: see text] is (weakly) enabling in [Formula: see text], then [Formula: see text] is not necessarily (weakly) enabling in [Formula: see text] while [Formula: see text] is (weakly) enabling in [Formula: see text]. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

Some results about proper overrings of pseudo-valuation domains

Let [Formula: see text] be a (unital) extension of (commutative) rings. We say that [Formula: see text] is a maximal non-quasi-local (respectively, non-PVD) subring of [Formula: see text] if [Formula: see text] is not quasi-local (respectively, PVD) and each subring of [Formula: see text] properly containing [Formula: see text] is quasi-local (respectively, PVD). The aim of this paper is to study this kind of ring extensions and to investigate the structure of the intermediate rings between [Formula: see text] and [Formula: see text].

Geminal Bis- and Tristriazoles: Long-known but still Uncommon Heterocyclic Entities

Geminal bis- and tristriazoles belong to a rare and mainly overlooked class of organic compounds with interesting properties. This review summarizes the methods for their preparation. Moreover, applications of these compounds are described, ranging from ring extensions to their use as reactive intermediates for the efficient construction of heterocycles. 1 Introduction 2 Bistriazoles 2.1 Bistriazoles (Tetrahedral Carbon) 2.2 Bistriazoles (Trigonal Carbon) 3 Tristriazoles 4 Conclusion and Prospectives

Repositioning Antitubercular 6-Nitro-2,3-dihydroimidazo[2,1-b][1,3]oxazoles for Neglected Tropical Diseases: Structure–Activity Studies on a Preclinical Candidate for Visceral Leishmaniasis

6-Nitro-2,3-dihydroimidazo[2,1-b][1,3]oxazole derivatives were initially studied for tuberculosis within a backup program for the clinical trial agent pretomanid (PA-824). Phenotypic screening of representative examples against kinetoplastid diseases unexpectedly led to the identification of DNDI-VL-2098 as a potential first-in-class drug candidate for visceral leishmaniasis (VL). Additional work was then conducted to delineate its essential structural features, aiming to improve solubility and safety without compromising activity against VL. While the 4-nitroimidazole portion was specifically required, several modifications to the aryloxy side chain were well-tolerated e.g., exchange of the linking oxygen for nitrogen (or piperazine), biaryl extension, and replacement of phenyl rings by pyridine. Several less lipophilic analogues displayed improved aqueous solubility, particularly at low pH, although stability toward liver microsomes was highly variable. Upon evaluation in a mouse model of acute Leishmania donovani infection, one phenylpyridine derivative (37) stood out, providing efficacy surpassing that of the original preclinical lead.

ON QUASI-COMMUTATIVE RINGS

We study the structure of central elements in relation with polynomial rings and introduce quasi-commutative as a generalization of commutative rings. The Jacobson radical of the polynomial ring over a quasi-commutative ring is shown to coincide with the set of all nilpotent polynomials; and locally finite quasi-commutative rings are shown to be commutative. We also provide several sorts of examples by showing the relations between quasi-commutative rings and other ring properties which have roles in ring theory. We examine next various sorts of ring extensions of quasi-commutative rings.

Drained residual shear strength at effective normal stresses relevant to soil slope stability analyses

An extensive torsional ring shear testing program has been conducted to measure the drained residual shear strength of soils at the wide range of effective normal stress (10 to 700kPa) usually mobilized in reactivated and first-time landslides. Soils, mudstones and shales of different plasticity and gradation were tested in the program. The effects of the change in nonlinearity of shear strength envelope over the utilized normal stress ranges on slope stability analyses were investigated. Using this data, new empirical residual shear strength correlations were developed as a function of soil index parameters and wide range of effective normal stresses. In essence, the correlations are presented as revised versions of those previously developed for a limited number of normal stresses utilizing the same soil index parameters. Comparisons were made with a considerable amount of back-calculated shear strength data reported in the literature for reactivated landslides as well as results predicted from existing shear strength correlations to verify the increased suitability of the new correlations for use in slope stability analyses. A numerical expression was also introduced to express the residual shear strength correlations for direct incorporation in slope stability software.

On the Jacobson Radical of Skew Polynomial Extensions of Rings Satisfying a Polynomial Identity

Let R be a ring satisfying a polynomial identity, and let D be a derivation of R. We consider the Jacobson radical of the skew polynomial ring R[x; D] with coefficients in R and with respect to D, and show that J(R[x; D]) ∩ R is a nil D-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when R is commutative.

Accounting for littoral primary production by periphyton shifts a highly humic boreal lake towards net autotrophy

Abstract The prevailing view that many humic lakes are net heterotrophic is commonly based on pelagic measurements alone. Poor light conditions in humic lakes are assumed to constrain littoral primary production (PP), such that the littoral zone has been considered an insignificant contributor to whole‐lake PP. However, that assumption is based on models and inferences from pelagic processes which do not take littoral zone structure into account. Many lakes have an extensive ring of aquatic vegetation lying near the water surface, which provides substratum for epiphytic algae under well‐illuminated conditions. We measured both pelagic and littoral PP and community respiration (CR) in Mekkojärvi, a small, highly humic headwater lake, in southern Finland throughout the open water season in 2012. We used a 14C incorporation technique to measure pelagic PP, while littoral PP was determined using changes in dissolved inorganic carbon concentrations during in situ incubations. We then estimated whole‐lake PP and CR for both the littoral and pelagic zones. We found that littoral PP usually dominated whole‐lake PP, contributing >90% to total PP in summer. A mean pelagic production to respiration ratio (GPP : CR) of 0.4 clearly indicated strong net heterotrophy, but a value of 2.2 for the littoral zone indicated strong autotrophy. For both habitats combined, the mean whole lake GPP : CR was 1.6, indicating net autotrophy. We suggest that littoral PP can contribute significantly to whole‐lake PP even in highly humic lakes, and that the littoral contribution can shift some lakes which have been considered net heterotrophic to metabolic balance or even net autotrophy during the ice‐free period. Small humic lakes like Mekkojärvi with rings of littoral vegetation are widespread, especially in the boreal zone, and at least for similar lakes, evaluating lake metabolism from pelagic measurements alone is likely to be misleading.

The direct summand conjecture for some bigenerated extensions and an asymptotic version of Koh’s conjecture

This article deals with two different problems in commutative algebra. In the first part we give a proof of the direct summand conjecture for module-finite extension rings of mixed characteristic $$R\subset S$$ satisfying the following hypotheses: the base ring R is a unique factorization domain of mixed characteristic zero. We assume that S is generated by two elements which satisfy, either radical quadratic equations, or general quadratic equations under certain arithmetical restrictions. In the second part of this article we discuss an asymptotic version of Koh’s conjecture. We give a model theoretical proof using “non-standard methods”.

Multiple Finger Extensor Tendon Dislocations in Systemic Lupus Erythematosus with Unique Trigger Phenomena

Dislocation of extensor tendons is the most frequently seen complaint for suffers of rheumatoid arthritis. Here, we present a rare clinical case of multiple finger extensor tendon dislocations in systemic lupus erythematosus (SLE). Upon physical examination, active extension of middle, ring and little fingers were impossible at the MP joint. A trigger phenomenon was generated when these extensors were reduced in passive extension. The procedure was performed under local anesthesia. A satisfactory clinical outcome was achieved with sagittal band reconstruction and reinforcement for the site using juncturae tendinum. Surgical intervention under the local anesthesia allowed for extensive gathering of information about repair site strength and dynamic stability.