Ring Extensions Research Articles

Commutative ring extensions defined by perfect-like conditions

UDC 512.5 In 2005, Enochs, Jenda, and López-Romos extended the notion of perfect rings to n -perfect rings such that a ring is n -perfect if every flat module has projective dimension less or equal than n . Later, Jhilal and Mahdou defined a commutative unital ring R to be strongly n -perfect if any R -module of flat dimension less or equal than n has a projective dimension less or equal than n . Recently Purkait defined a ring R to be n -semiperfect if R ¯ = R / R a d ( R ) is semisimple and n -potents lift modulo R a d ( R ) . We study of three classes of rings, namely, n -perfect, strongly n -perfect, and n -semiperfect rings. We investigate these notions in several ring-theoretic structures with an aim of construction of new original families of examples satisfying the indicated properties and subject to various ring-theoretic properties.

A Note on Star-Pr"ufer Extensions

Let ★ be a star operation on a ring extension R \subseteq S . The ring extension R \subseteq S is said to be a ★-Pr¨ufer if R[p] \subseteq S is a Pr¨ufer extension for each ★-prime ideal p of R. We study properties of ★-Pr¨ufer extensions. In particular, we investigate the transfer of star-Pr¨ufer properties from the extension R \subseteq S to the extension R[X] \subseteq S [X] of polynomial rings, where X is an indeterminate over S .

Spatiotemporal expansion modes of urban areas on the Loess Plateau from 1992 to 2021 based on nighttime light images

Long-term nighttime light data can effectively reflect the spatiotemporal evolution pattern of urban areas. Based on nighttime light data obtained from 1992 to 2021, the spatiotemporal expansion modes of urban areas on the Loess Plateau were investigated by remote sensing image band fusion processing, geospatial analysis, statistics and classification methods. Results show that the characteristics of the spatiotemporal expansion of urban areas were retained in the extraction boundaries. From the perspective of time scales, the urban areas have grown tripled to 11,481 km2. Four significant instances of urban areas increase occurred, with one large increase occurring almost every ten years. From the perspective of spatial scales, urban area expansion has two important features. First, the expansion speed of urban areas, which is based on three indicators, has large spatial dissimilarities. Five modes can be divided based on the expansion velocity of 44 cities/counties. Second, urban development increases uniquely, and can be described based on three expansion modes: Ring Extension, which mainly occurs in wide and flat basins of the main rivers; Linear Extension primarily arises on both banks of rivers in hilly and gully areas; and Point Extension mainly occurs in areas away from the main rivers. Our results highlight that the spatiotemporal expansion modes of urban areas on the Loess Plateau retain the significant influence of the geological environment, which can provide scientific guidance for the sustainable development of human-land relationships.

Local Nullstellensatz over Commutative Ground Rings

It is shown that a local Nullstellensatz holds over an arbitrary commutative ring A (with identity 1 ≠ 0); specifically, if B = A[x1, . . . , xn] is a finitely generated extension ring of A and N is a maximal ideal in B, then NBN = (N ∩ A, x1 − c1, . . . , xn − cn)BN for some c1, . . . , cn ∈ BN .

Adjacency-like conditions and induced ideal graphs

In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We examine the interplay between the multiplicative ideal theory and the graph theoretic structure of the associated graph. Behavior of these graphs under standard ring extensions are studied, and in conjunction with the theory, some classical results and connections are made.

Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

Radiation dose optimization for photon-counting CT coronary artery calcium scoring for different patient sizes: a dynamic phantom study

PurposeTo systematically assess the radiation dose reduction potential of coronary artery calcium (CAC) assessments with photon-counting computed tomography (PCCT) by changing the tube potential for different patient sizes with a dynamic phantom.MethodsA hollow artery, containing three calcifications of different densities, was translated at velocities corresponding to 0, < 60, 60–75, and > 75 beats per minute within an anthropomorphic phantom. Extension rings were used to simulate average- and large -sized patients. PCCT scans were made with the reference clinical protocol (tube potential of 120 kilovolt (kV)), and with 70, 90, Sn100, Sn140, and 140 kV at identical image quality levels. All acquisitions were reconstructed at a virtual monoenergetic energy level of 70 keV. For each calcification, Agatston scores and contrast-to-noise ratios (CNR) were determined, and compared to the reference with Wilcoxon signed-rank tests, with p < 0.05 indicating significant differences.ResultsA decrease in radiation dose (22%) was achieved at Sn100 kV for the average-sized phantom. For the large phantom, Sn100 and Sn140 kV resulted in a decrease in radiation doses of 19% and 3%, respectively. Irrespective of CAC density, Sn100 and 140 kVp did not result in significantly different CNR. Only at Sn100 kV were there no significant differences in Agatston scores for all CAC densities, heart rates, and phantom sizes.ConclusionPCCT at tube voltage of 100 kV with added tin filtration and reconstructed at 70 keV enables a ≥ 19% dose reduction compared to 120 kV, independent of phantom size, CAC density, and heart rate.Key Points• Photon-counting CT allows for reduced radiation dose acquisitions (up to 19%) for coronary calcium assessment by reducing tube voltage while reconstructing at a normal monoE level of 70 keV.• Tube voltage reduction is possible for medium and large patient sizes, without affecting the Agatston score outcome.

Variation in nonadiabaticity and dual fluorescence causality by extension with aromatic rings in tropolone

Photophysical aspects of tropolone upon extension with aromatic rings are studied theoretically. Tropolone possesses three energetically close lying singlet excited states (S 1 (1 π π ∗ ), S 2 (2 π π ∗ ) and S 3 (1 n π ∗ )), resulting in a rapid internal conversion among these states upon photoexcitation. The excited-state energetics and decay dynamics of benzotropolone are similar to the tropolone molecule. On the other hand, upon extending the tropolone with naphthalene moiety (naphthatropolone), the molecule exhibits relatively well-separated electronic states, thus displaying significantly slower photorelaxation compared to the tropolone molecule. Both benzotropolone and naphthatropolone exhibit the dual emission feature, which is absent in the parent tropolone due to the symmetrical double-well potential of S 1 . • Loss of near-degeneracy upon ring extension in tropolone. • Competing nonadiabatic relaxation and ESIPT. • O-H vibrations facilitates rapid internal conversion. • Dual emission features in extended tropolones.

Classes of trivial ring extensions and amalgamations subject to pseudo-almost valuation condition

Abstract This paper studies the transfer of pseudo-almost valuation property (PAVR property for short) to various context of commutative ring extensions such as power series ring, trivial ring extension and amalgamation. Our work is motivated by an attempt to generate new original classes of rings satisfying this property. The obtained results are backed with new and illustrative examples arising as trivial ring extensions, amalgamations and pullback constructions.

ON NIL SKEW GENERALIZED POWER SERIES REFLEXIVE RINGS

Let $R$ be a ring and $(S, \leq)$ a strictly ordered monoid. In this paper, we deal with a new approaches to reflexive property for rings by using nilpotent elements. In this direction we introduce the notions of $(S, \omega)$-reflexive and $(S, \omega)$-$nil$-reflexive. Examples are given that, $(S, \omega)$-$nil$-reflexive is not $(S, \omega)$-reflexive. Under some suitable conditions, we proved that, if $R$ is a right $APP$-ring, then $R$ is $(S, \omega)$-reflexive and $R$ be a semiprime ring with the $ACC$ on left annihilator ideals, $(S, \leq)$ an $a.n.u.p.$-monoid, then $R$ is $(S, \omega)$-reflexive. Also, we proved that, $R$ is $(S, \omega)$-$nil$-reflexive if and only if $R/I$ is $(S, \overline{\omega})$-$nil$-reflexive, $R$ is $(S, \omega)$-$nil$-reflexive if and only if $T_{n}(R)$ is $(S, \omega)$-$nil$-reflexive and we will show that, if $R$ is a right Noetherian ring, then $R$ is $(S, \omega)$-$nil$-reflexive. Moreover, we investigate ring extensions which have roles in ring theory.

The structure of matrix polynomial algebras

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix-variables over a ground polynomial ring in not necessary commuting variables. These matrix subalgebras of full matrix rings over polynomial rings show up in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, which are interesting noncommutative algebras arising from the study of $\mathrm{Gl}_{n}$-varieties.

Essential finite generation of extensions of valuation rings

AbstractGiven a generically finite local extension of valuation rings , the question of whether W is the localization of a finitely generated V‐algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when W is essentially of finite type over V in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic 0 using embedded resolution of singularities. In this paper, we prove Knaf's conjecture in full generality.

Hepatic lesion detectability in abdomen computed tomography: Investigation in low kVp single energy and low keV virtual monochromatic images generated from dual energy computed tomography using task-based image quality assessment in phantom

Background: Low tube potential single-energy (SE) and virtual monochromatic image (VMI) dual energy (DE) abdomen CT images both improve the hepatic lesion detection by increasing the liver lesion contrast on images. Objectives: To study the performance of low kVp single energy and low keV virtual monochromatic images (VMI) generated from dual energy acquisition to detect the hepatic lesion on abdominal CT imaging in phantom. Materials and methods: The anthropomorphic liver with nodule inserted phantom with extension rings simulating the small, medium, and large patient was scanned under the SECT acquisition by varying the kVp from 70-120 kVp and for DECT acquisitions, three kVp combinations (80/-,90/-, and 100/Sn150-kVp). The series of 40-,50-,60-, and 70-keV VMI were generated from DECT data set. All images were used to assess the task-based image quality; task transfer function (TTF), noise power spectrum (NPS), and detectability index (d’) with the diagnostic task to detect 15 mm diameter hyperattenuating hepatic lesion. Results: The result showed that the TTF was higher in low kVp SECT while the lesion contrast was higher at low keV VMI -DECT. The noise magnitude remained constant for all kVp values in SECT, but it was dramatically increased as decreased the energy level from 70- to 40-keV in VMI-DECT. The fav of NPS shifted to higher frequency when increasing the kVp and when increasing the energy level of VMI. The obtained d’ was highest in low kVp SECT at 70-or 80-kVp. Conclusion: The low kVp SECT provided the highest d’ than that in low keV VM image from DECT in all phantom sizes where the highest d’ was found at 70 kVp-SECT for small and medium phantom and at 80 kVp for large phantom. For SECT, reduced the kVp to 70- or 80-kVp improved the detectability index. The kVp combination in DECT impacts to the d’ of VMI; at small phantom, the highest d’ for each keV VMI was found at 80/Sn150 kVp acquisition and at larger phantom size, higher kVp on tube “A” is more favored.

On the diameter of the zero-divisor graph over skew PBW extensions

The aim of this paper is to investigate the interplay between the algebraic properties of a skew Poincaré–Birkhoff–Witt extesion ring [Formula: see text] and the graph-theoretic properties of its zero-divisor graph. We are interested in studying the diameter of the zero-divisor graph of skew PBW extension rings. Among other results, we give a complete characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text].

Non-unital Ore extensions

We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;\delta ]$, under the hypothesis that $R$ is $s$-unital and $\ker (\delta )$ contains a non-zero idempotent. This

Acene enlargement for absorption red-shifting and photosensitization enhancement of photosensitizers with aggregation-induced emission.

Photosensitizers with aggregation-induced emission (AIE PSs) were widely explored in photodynamic therapy. Numerous acceptors but few donors were reported to design AIE PSs. In this study, we developed a new kind of donor that can improve the comprehensive performance of AIE PSs by expanding the π extension of aromatic rings at the end of the triphenylamine group through acene enlargement. The absorption and fluorescence peaks of anthryl-substituted AIE PS are red-shifted by 29 nm and 42 nm; the photosensitization efficiency is enhanced by 1.16 times; the AIE factor is 86.1 and the fluorescence quantum yield is 9.3%. We also demonstrated that the anthryl-based AIE PS can image and ablate cancer cells well both in vitro and in vivo. The anthryl-triphenylamine donor provides an excellent option to design donor-acceptor AIE PSs with high comprehensive performance.

IMMEDIATE EXTENSIONS OF VALUATION RINGS AND

We describe the immediate extensions of a one dimensional valuation ring V which could be embedded in some separation of a ultrapower of V with respect to a certain ultrafilter. For such extensions, a kind of Artin’s approximation holds.

Left Nil Zero Semicommutative rings

This paper introduces a class of rings called left nil zero semicommutative rings ( LNZS rings ), wherein a ring R is said to be LNZS if the left annihilator of every nilpotent element of R is an ideal of R. It is observed that reduced rings are LNZS but not the other way around. So, this paper provides some conditions for an LNZS ring to be reduced and among other results, it is proved that R is reduced if and only if the ring of upper triangular matrices over R is LNZS. Furthermore, it is shown that the polynomial ring over an LNZS may not be LNZS and so is the case of the skew polynomial over an LNZS ring. Therefore, this paper investigates the LNZS property over the polynomial extension and skew polynomial extension of an LNZS ring with some additional conditions.

Application of extension rings in thermography for electronic circuits imaging

ABSTRACT Thermographic cameras are often used in areas, where resolutions below 100 µm per pixel are required, as for example in electronics. Such resolutions can be obtained using dedicated lenses. As an alternative, some manufacturers are offering extension rings to be added between the lens and the camera housing, to increase the final image spatial resolution. To test the latter solution and its impact on thermographic temperature measurements, a set of aluminium extension rings was manufactured and mounted in different combinations between an InSb cooled thermographic camera and its lens. Next, the camera, a blackbody IR source, a thick-film resistor on alumina substrate, a microscope reticle calibration glass, and two integrated test circuits were used to investigate the impact of the rings on the temperature measurement accuracy and the achieved spatial resolution. It was demonstrated, that extension rings are a very useful tool in thermography, as long as their impact on the measurement system is taken into account, considering numerical aperture reduction, rings heating by the operating camera, and the cold stop related vignetting. The impact will greatly depend on the specific design of the camera to be used and its lens configuration.

BIG COHEN–MACAULAY TEST IDEALS IN EQUAL CHARACTERISTIC ZERO VIA ULTRAPRODUCTS

AbstractUtilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra $\mathcal {B}(R)$ over a local domain R essentially of finite type over $\mathbb {C}$ . We show that if R is normal and $\Delta $ is an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier, then the BCM test ideal $\tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ with respect to $\widehat {\mathcal {B}(R)}$ coincides with the multiplier ideal $\mathcal {J}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ , where $\widehat {R}$ and $\widehat {\mathcal {B}(R)}$ are the $\mathfrak {m}$ -adic completions of R and $\mathcal {B}(R)$ , respectively, and $\widehat {\Delta }$ is the flat pullback of $\Delta $ by the canonical morphism $\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$ . As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.