Ring Extensions Research Articles

The implementation of algebraic complex lookup tables over non-chain galois ring extensions and henon map in multimedia security

Abstract Image encryption is crucial for web-based data storage and transmission. Complex algebraic structures play a vital role in providing unique features and binary operations. However, current algebraic-based techniques face challenges due to limited key space. To tackle this issue, our study uniquely connects the algebraic structures with a chaotic map. The study introduces a complex non-chain Galois ring structure and a 12-bit substitution box for image substitution. An affine map is utilized to permute image pixels, and the 12-bit substitution box is uniquely mapped to a Galois field for encryption. A two-dimensional Henon map is employed to generate different keys for the XOR operation, resulting in an encrypted image. The resilience of the scheme against various attacks is evaluated using statistical, differential, and quality measures, showcasing its effectiveness against well-known attacks.

The Local Picard Group of a Ring Extension

Given an integral domain D and a D-algebra R, we introduce the local Picard group LPic(R,D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{LPic}(R,D)$$\\end{document} as the quotient between the Picard group Pic(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pic}(R)$$\\end{document} and the canonical image of Pic(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pic}(D)$$\\end{document} in Pic(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pic}(R)$$\\end{document}, and its subgroup LPicu(R,D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{LPic}_u(R,D)$$\\end{document} generated by the the integral ideals of R that are unitary with respect to D. We show that, when D⊆R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D\\subseteq R$$\\end{document} is a ring extension that satisfies certain properties (for example, when R is the ring of polynomial D[X] or the ring of integer-valued polynomials Int(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Int}(D)$$\\end{document}), it is possible to decompose LPic(R,D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{LPic}(R,D)$$\\end{document} as the direct sum ⨁LPic(RT,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bigoplus \ extrm{LPic}(RT,T)$$\\end{document}, where T ranges in a Jaffard family of D. We also study under what hypothesis this isomorphism holds for pre-Jaffard families of D.

Characterizations of reversible rings relating k-potents

This article embodies a ring theoretic property that preserves the reversibility at non zero k-potents. The structure of k-potents in k-reversible and pseudo k-reversible rings are studied in connection with various kinds of ring extensions. A ring R is said to be k-reversible if and only if ab ∈ K(R) for a,b ∈ R implies ab = ba; and a ring R is called pseudo k-reversible if 0 ̸= ab ∈ K(R) for a,b ∈ K(R) implies ab = ba, where K(R) is the set of all k-potents in R. It is proved that k-reversible rings are pseudo k-reversible but the converse is not true in general. Under certain conditions the polynomial rings of pseudo k-reversible inherit the character of pseudo k-reversible rings.

Noetherian rings of composite generalized power series

Abstract Let A ⊆ B A\subseteq B be an extension of commutative rings with identity, ( S , ≤ ) \left(S,\le ) a nonzero strictly ordered monoid, and S * = S \ { 0 } {S}^{* }\left=S\backslash \left\{0\right\} . Let A + 〚 B S * , ≤ 〛 = { f ∈ 〚 B S , ≤ 〛 ∣ f ( 0 ) ∈ A } A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] =\{f\in [\kern-2pt[ {B}^{S,\le }]\kern-2pt] \hspace{0.33em}| \hspace{0.33em}f\left(0)\in A\} . In this study, we determine when the ring A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring. We prove that when S S is a strict monoid, if A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring, then A A is a Noetherian ring, B B is a finitely generated A A -module, and S S is finitely generated. We also show that if B B is a finitely generated A A -module over a Noetherian ring A A and ( S , ≤ ) \left(S,\le ) is a positive strictly ordered monoid, which is finitely generated, then A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring.

Hurwitz series rings satisfying a zero divisor property

In this paper, we study zero divisors in the Hurwitz series rings and the Hurwitz polynomial rings over general non-commutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties of (the Hurwitz series) Armendariz rings. We show that for a semiprime Armendariz of the Hurwitz series type (so reduced) ring [Formula: see text] with [Formula: see text] on annihilator ideals, HR (the Hurwitz series ring with coefficients over [Formula: see text]) has finitely many minimal prime ideals, say [Formula: see text] such that [Formula: see text] and [Formula: see text] for some minimal prime ideal [Formula: see text] of [Formula: see text] for all [Formula: see text], where [Formula: see text] are all minimal prime ideals of [Formula: see text]. Additionally, we construct various types of (the Hurwitz series) Armendariz rings and demonstrate that the polynomial ring extension preserves the Armendarizness of the Hurwitz series as the Armendarizness.

On generalized Star-GCD extensions

– Let ⋆ be a star operation on a ring extension R ⊆ S . The ring extension is said to be a generalized ⋆ -greatest common divisor (G- ⋆ GCD) extension if the intersection of two ⋆ -invertible ideals of R is ⋆ -invertible. We investigate properties of G- ⋆ GCD extensions. We give special attention to polynomial rings and Nagata rings.

Stacking interaction and opto-electronic properties of star-shaped benzotrithiophene and its extended derivatives

Benzotrithiophene and its derivatives obtained by fusing the aromatic rings at the peripheral sites are modelled using density functional methods for exploring their optoelectronic properties. The studies showed that absorption maximum shifted towards higher wavelengths with the extension of rings, irrespective of the functional used. The above molecules adopt anti-parallel orientations in their stacked dimer. The analysis of the stacked dimers revealed that they interact via van der Waals forces. The charge transport properties are found to vary with the orientation of the molecules. The studies showed that the designed molecules act as electron transporters in their anti-parallel orientation.

Reynolds number effect of a vortex ring impinging on a concave hemi-cylindrical shell

Experimental investigations were conducted on a single vortex ring impinging on a concave hemi-cylindrical shell with Dm/De = 2 at different Reynolds numbers. Vortex rings with five different Reynolds numbers were generated for experimental studies, i.e., Re = 750, 1500, 3000, 5000, and 7000. The planar laser-induced fluorescence visualizations and two-dimensional particle image velocimetry measurements were used in the experiment. The vorticity field based on the Eulerian framework and the finite-time Lyapunov exponent (FTLE) field based on the Lagrangian framework were used to identify the dynamic processes of vortex rings, respectively. The results show that as the vortex rings impinge on concave surfaces from Re = 750 to Re = 7000, the extension of the main vortex ring in the straight-edged direction is larger than that in the concave direction, and the instability of the vortex ring is promoted. While the Reynolds number is increasing, the vortex ring deformation becomes larger, and the overall vortex ring cross section becomes smaller, leading to a larger attenuation of the vortex ring rotation. Calculations performed by the FTLE field were used to derive the Lagrangian coherent structure to analyze the boundaries of the vortex ring motion process, clearly observe the shape of the secondary vortex connecting segments, and verify the speculation by the vortex ring trajectory identification results. Finally, a dynamic model of vortex rings impinging a concave surface was proposed, and the inference of the experimental process was explained by the model.

Computing the closure of a support

When E is an R-module over a commutative unital ring R, the Zariski closure of its support is of the form V ( O ( E ) ) where O ( E ) is a unique radical ideal. We give an explicit form of O ( E ) and study its behavior under various operations of algebra. Applications are given, in particular for ring extensions of commutative unital rings whose supports are closed. We provide some applications to crucial and critical ideals of ring extensions.

Two absorbing factorization formal power series rings

Let R be a commutative ring with identity. A proper ideal I of R is said to be 2-absorbing if whenever xyz ∈ I for some elements x , y , z ∈ R , then either xy ∈ I or xz ∈ I or yz ∈ I . The ring R is called a two-absorbing factorization ring (TAF-ring) if every proper ideal of R has a two absorbing-factorization that is every ideal is a product of 2-absorbing ideals. In this note, we characterize commutative rings R (respectively, commutative ring extensions A ⊂ B ) for which the ring of formal power series R [ [ X ] ] (respectively, the ring A + XB [ [ X ] ] ) is a TAF-ring.

Right Central CNZ Property Skewed by Ring Endomorphisms

The concept of the reversible ring property concerning nilpotent elements was introduced by A.M. Abdul-Jabbar and C. A. Ahmed, who introduced the concept of commutativity of nilpotent elements at zero, termed as a CNZ ring, as an extension of reversible rings. In this paper, we extend the CNZ property through the influence of a central ring endomorphism alpha , introducing a new type of ring called a right alpha -skew central CNZ ring. This concept not only expands upon CNZ rings but also serves as a generalization of right alpha -skew central reversible rings. We explore various properties of these rings and delve into extensions of right alpha -skew central CNZ rings, along with examining several established results, which emerge as corollaries of our findings.

Differentiably simple rings and ring extensions defined by p-basis

We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit p-basis. These extensions are called Galois extensions of exponent one. For such an extension A⊂C, we introduce an A-scheme, called the Yuan scheme, which parametrizes subextensions A⊂B⊂C such that B⊂C is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers.

On ring extensions pinched at the integral closure

On ring extensions pinched at the integral closure

On S-2-Prime Ideals of Commutative Rings

Prime ideals and their generalizations are crucial in numerous research areas, particularly in commutative algebra. The concept of generalization of prime ideals begins with the study of weakly prime ideals. Since then, subsequent works aimed at expanding this concept into more generalized forms. Among these, S-prime ideals and 2-prime ideals have reaped attention recently. This paper aims to characterize S-2-prime ideals, which serve as a generalization encompassing both 2-prime ideals and S-prime ideals. To accomplish this objective, we construct an ideal which distinct from a multiplicatively closed subset with the help of commutative rings. We investigate the localization and the S-2-prime avoidance lemma in commutative rings. Furthermore, we explore the properties of this class of ideals in trivial ring extensions and amalgamated algebras along an ideal. We delve into S-properties for compactly packedness, compactly 2-packedness and coprimely packedness in trivial ring extentions. Moreover, this notion of ideals helps us to indicate that many results stated in S-prime ideals and 2-prime ideals can be readily expanded to the framework of S-2-prime ideals. Supporting examples also highlight a significant distinction between S-2-prime ideals and stated ideals.

Normal Completions of Toric Varieties Over Rank One Valuation Rings and Completions of Γ-Admissible Fans

Abstract We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $\Gamma $ is discrete or divisible then no base-change is needed. We give explicit examples that show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $\Gamma $-admissible fans admit $\Gamma $-admissible completions. In order to show this we prove a combinatorial analog of Noetherian reduction, which we believe will be of independent interest.

Patient-derived PixelPrint phantoms for evaluating clinical imaging performance of a deep learning CT reconstruction algorithm.

Objective
Deep learning reconstruction (DLR) algorithms exhibit object-dependent resolution and noise performance. Thus, traditional geometric CT phantoms cannot fully capture the clinical imaging performance of DLR. This study uses a patient-derived 3D-printed PixelPrint lung phantom to evaluate a commercial DLR algorithm across a wide range of radiation dose levels.

Method
The lung phantom used in this study is based on a patient chest CT scan containing ground glass opacities and was fabricated using PixelPrint 3D-printing technology. The phantom was placed inside two different size extension rings to mimic a small- and medium-sized patient and was scanned on a conventional CT scanner at exposures between 0.5 and 20 mGy. Each scan was reconstructed using filtered back projection (FBP), iterative reconstruction, and DLR at five levels of denoising. Image noise, contrast to noise ratio (CNR), root mean squared error (RMSE), structural similarity index (SSIM), and multi-scale SSIM (MS SSIM) were calculated for each image.

Results
DLR demonstrated superior performance compared to FBP and iterative reconstruction for all measured metrics in both phantom sizes, with better performance for more aggressive denoising levels. DLR was estimated to reduce dose by 25-83% in the small phantom and by 50-83% in the medium phantom without decreasing image quality for any of the metrics measured in this study. These dose reduction estimates are more conservative compared to the estimates obtained when only considering noise and CNR. 

Conclusion
DLR has the capability of producing diagnostic image quality at up to 83% lower radiation dose, which can improve the clinical utility and viability of lower dose CT scans. Furthermore, the PixelPrint phantom used in this study offers an improved testing environment with more realistic tissue structures compared to traditional CT phantoms, allowing for structure-based image quality evaluation beyond noise and contrast-based assessments.

Nonnil-FP-injective and nonnil-FP-projective dimensions and nonnil-semihereditary rings

In this paper, we introduce the concept of nonnil-FP-injective dimension for both modules and rings. We explore the characterization of strongly ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-rings that have a nonnil-FP-injective dimension of at most one. We demonstrate that, for a nonnil-coherent, strongly ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-ring R, the nonnil-FP-injective dimension of R corresponds to the supremum of the ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-projective dimensions of specific families of R-modules. We also define self-nonnil-injective rings as ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-rings that act as nonnil semi-injective modules over themselves and establish the equivalence between a strongly ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-ring R being ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-von Neumann regular and R being both nonnil-coherent and self-nonnil semi-injective. Furthermore, we extend the notion of semihereditary rings to ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-rings, coining the term ‘nonnil-semihereditary’ to describe rings where every finitely generated nonnil ideal is u-ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-projective. We provide several characterizations of nonnil-semihereditary rings through various conceptual lenses. Our study also includes an investigation of the transfer of the nonnil-semihereditary property in trivial ring extensions. Additionally, we define the nonnil-FP-projective dimension for modules and rings, showing that for any strongly ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-ring, a nonnil-FP-projective dimension of zero is indicative of the ring being nonnil-Noetherian. We also ascertain that, for a strongly ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-ring R, its nonnil-FP-projective dimension is the supremum of the NFP-projective dimensions across different families of R-modules. Lastly, we provide numerous examples to illustrate our results.

The ring A[X,Y;λ] and the SFT property

Let A be a ring and λ : R ≥ 0 → R ≥ 0 be an increasing nonzero function. In this paper, we show that if A is a Noetherian ring with characteristic n ≠ 0 and lim k → + ∞ λ ( k + 1 ) λ ( k ) = + ∞ , then A [ X , Y ; λ ] is an SFT ring. This result allows us to construct a nonNoetherian SFT ring which has some Noetherian completion. Also, we use the composite ring extension to construct many examples of such rings. Many examples are provided.

Dual-source photon-counting CT: impact of residual cross-scatter on quantitative spectral results.

Dual-source photon-counting CT combines the high temporal resolution and high pitch of dual-source CT with the material quantification capabilities of photon-counting CT. It, however, results in cross-scatter that increases in severity with increased patient size and collimation. This cross-scatter must be corrected to ensure the removal of scatter artifacts and improve quantitative accuracy. To evaluate residual cross-scatter of a first-generation dual-source photon-counting CT and the effect of phantom size, collimation, and radiation dose, a phantom was scanned in single- and dual-source modes with and without its extension ring at three collimations and three radiation doses. Virtual monoenergetic images (VMI) at 50 keV, VMI 150 keV, and iodine density maps were reconstructed to determine variation between acquisition parameters in single- and dual-source modes. Additionally, differences relative to single-source acquisitions and to single-source and small collimation acquisitions were calculated to reflect residual cross-scatter with and without matched collimation. At VMI 50 keV, inserts exhibited accuracy and similar variation between single- and dual-source modes, averaging 5.4 ± 2.6 and 6.2 ± 2.5 HU, respectively, across phantom size, collimation, and radiation dose. Differences relative to single-source measured 5.1 ± 8.5 and 0.4 ± 4.2 HU while differences relative to single-source and small collimation acquisitions were 6.4 ± 10.8 HU and -0.5 ± 3.9 HU for VMI 50 and 150 keV, respectively. This minimal residual cross-scatter increases confidence in the quantitative accuracy of spectral results necessary for clinical applications of dual-source photon-counting CT with motion, such as cardiac imaging.

Quasi-hereditary rings are closed under taking block extensions

In this paper, we give a sufficient condition for Morita context rings to be quasi-hereditary. As an application, we show that each block extension of a quasi-hereditary ring is also quasi-hereditary.