Regular Ring Research Articles

Branched covers and matrix factorizations

AbstractLet be a regular local ring and a non‐zero element of . A theorem due to Knörrer states that there are finitely many isomorphism classes of maximal Cohen–Macaulay (CM) ‐modules if and only if the same is true for the double branched cover of , that is, the hypersurface ring which is defined by in . We consider an analogue of this statement in the case of the hypersurface ring defined instead by for . In particular, we show that this hypersurface, which we refer to as the ‐fold branched cover of , has finite CM representation type if and only if, up to isomorphism, there are only finitely many indecomposable matrix factorizations of with factors. As a result, we give a complete list of polynomials with this property in characteristic zero. Furthermore, we show that reduced ‐fold matrix factorizations of correspond to Ulrich modules over the ‐fold branched cover of .

Expected Resurgence of Ideals Defining Gorenstein Rings

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence and thus satisfy the stable Harbourne conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided that its symbolic powers are given by saturations with the maximal ideal. Although this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is Noetherian.

Operators on regular rings of Leavitt path algebras

In [8, Theorem 1], Jain and Prasad obtained a kind of symmetry of regular rings which is interesting and useful in the theory of shorted operators (cf. [9]). We show that this symmetry property indeed holds for endomorphism rings of Leavitt path algebras. Using this property, we analyze a (strong/weak) regular inverse of an element of the regular endomorphism ring A of the Leavitt path algebra L:= LK(E) (viewed as a right L-module). We also introduce some partial orders on the endomorphism ring A of the Leavitt path algebra L and investigate the behavior of regular elements in A.

Nanoscale Saturn Systems Based on C60/70 Bucky Ball and a Newly Designed [4]Cyclopara-1,2-diphenylethylene Hoop: A Strategy for Fullerene Encapsulation Release and Selective Recognition for C70.

A new carbonaceous nanohoop, [4]cyclopara-1,2-diphenylethylene ([4]CPDPE, composed by four 1,2-diphenylethylene units linked via the para of the phenyls), is designed together with two rational synthesis paths being proposed. The Saturn-like host-guest systems formed with the [4]CPDPE nanoring and fullerene C60/70 are explored using density functional theory calculations. The results evidence that the geometry mutual matching between [4]CPDPE and C60/70 is perfect, and the [4]CPDPE⊃C60/70 complexes could be formed spontaneously with high binding energies. Thermodynamic calculation results show that it essentially prefers to selectively recognize C70 over its smaller cousin C60. More interestingly, the [4]CPDPE nanoring could present the regular ring cylinder and the saddle shapes via configuration transformation between its all-trans form and all-cis form, so as to theoretically realize the fullerene encapsulation and release under photoirradiation. Furthermore, the 2:1 interaction structure ([4]CPDPE2⊃Dimer-C60) and properties are investigated. Additionally, the ultraviolet-visible (UV-vis) spectra are simulated, and host-guest noncovalent interaction (NCI) regions are investigated based on the electron density and reduced density gradient (RDG), which may be helpful for a deep understanding of the present designed systems in future.

On ϕ-(weak) global dimension

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called [Formula: see text]-u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On [Formula: see text]-exact sequence and [Formula: see text]-projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of [Formula: see text]-flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On [Formula: see text]-Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on [Formula: see text]-Dedekind rings and [Formula: see text]-Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly [Formula: see text]-flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the [Formula: see text]-injective dimension, [Formula: see text]-projective dimension and [Formula: see text]-flat dimension for modules, and also the [Formula: see text]-(weak) global dimension of rings. Then, using these dimensions, we characterize several [Formula: see text]-rings ([Formula: see text]-Prüfer, [Formula: see text]-chained, [Formula: see text]-von Neumann, etc). Finally, we study the [Formula: see text]-(weak) global dimension of the trivial ring extensions defined by some conditions.

Analysis of the response characteristics of a roadway wall under the impact of gas explosion

AbstractMine gas explosion causes serious failure (damage) to the roadway, but the research on the response characteristics of the impact load of gas explosion to the roadway wall are still very limited. In view of the shortcomings of the existing research, the LS‐DYNA software is used to establish the physical and mathematical models of gas explosion in a roadway, and the validation results show that the model is effective and reliable. Based on the model, changes of pressure, velocity, displacement of roadway walls and equivalent stress under the impact of gas explosions are measured. The response characteristics of the end and wall of roadway under the thermal impact of gas explosion are analyzed. The results show that the pressure on the closed end wall, the center and edge of the roadway is relatively large, and the wall failure is also relatively more serious. With the propagation of gas explosion, the overpressure on the closed end wall gradually attenuates, and the maximum overpressure region also retracts to the center. At the closed end, the explosion pressure is first loaded on the inner wall and gradually transferred to the outer wall. During the transfer, the pressure decays step by step, and the velocity of the measuring point at the closed end decays continuously. With the propagation of gas explosion, the displacement of each measuring point in the Z direction increases continuously. In the axial direction, the displacement near the center of the roadway is large, and then the displacement decreases in the form of regular rings. The impact load produced by the gas explosion is first loaded onto the closed end wall, resulting in wall deformation and equivalent stress. The study results can provide some theoretical basis and data support for the roadway wall design, and reduction of the wall failure.

A tri-band flexible antenna based on tri-band AMC reflector for gain enhancement and SAR reduction

A tri-band wearable antenna fed by coplanar waveguide (CWP) with an artificial magnetic conductor (AMC) reflector is presented in this paper. The radiation patch consists of three open inverted triangles for multiband, and the ground is defected for better impedance matching The AMC unit is composed of three concentric nested regular octagonal rings. And to improve the performance of gain, the unit achieves three reflection zero phases at 1.8, 2.45 and 3.5 GHz corresponding to the three bands of the antenna. The proposed antenna based on AMC has been fabricated and measured. The working frequency bands are 1.34–1.67 GHz, 2.33–2.47 GHz and 3.55–3.7 GHz. The measured gain has increased by 3.4, 1.8, 4 dBi at 1.8, 2.45 and 3.5 GHz, respectively. The scenario that the antenna is placed on the human arm is also considered and experimented for wearable applications. The results are in good agreement with the simulation data and verify that the antenna has good robustness. When the antenna is conformal with the human tissue, the specific absorption rate (SAR) is calculated and reduced by about 99.8% after adding AMC reflector. Hence the proposed antenna has excellent wearable safety and is compatible with LTE 1800, WLAN, WiMAX communication systems.

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

SOME RESULTS ON STRONGLY π-REGULAR RIN

In this paper we study the strongly - regular ring (for short st. -reg. rg.) and some properties also give some new results of st. -reg. rg. and its connection with other rings.

R⁎-algebras and C⁎-uniqueness of pre-C⁎-algebras

In 1967, Kadison raised the following question, if a self-adjoint operator algebra on a Hilbert space is finitely generated (algebraically) and each self-adjoint operator in it has finite spectrum, is it finite dimensional? In 2021, M. Mori connected the Kadison's question to von Neumann's concept of regular rings given in 1930's, and introduced a notion of R⁎-algebras. In this paper, we study R⁎-algebras, the Kadison's question, and a related question of uniqueness of C⁎-norm on pre-C⁎-algebras. Precisely, we study atomic R⁎-algebras, and find a maximal ultramatricial R⁎-subalgebra of an atomic R⁎-algebra, which properly contains the maximal purely atomic R⁎-subalgebra generated by all finite projections of the algebra. We prove that the set of all ⁎-algebras for which the Kadison's question has an affirmative answer is closed under direct sum, tensor product, and crossed product, but it is not closed under free product. From the aforementioned stable properties of the algebra in the Kadison's question, we find plentiful examples of ⁎-algebras for which the Kadison's question has an affirmative answer, for instance, the crossed product ⁎-algebra B⋊αG, where G is a locally finite group and B is one of the following algebras: an algebra of finite rank operators, an abelian algebra, a finite dimensional algebra, a twisted group algebra satisfying a certain condition, a rotation algebra, or a group measure space construction algebra. We characterize uniqueness of C⁎-norm on a pre-C⁎-algebras in terms of its ⁎-representations, weak containment of its ⁎-representations, ideals of its enveloping C⁎-algebra, or the primitive ideal space of the C⁎-algebra. We also prove that a unital pre-C⁎-algebra has a unique C⁎-norm if and only if, for a faithful ⁎-representation of the algebra, invertibility of a self-adjoint element of the algebra, as an operator on the representation Hilbert space, is independent of the choice of the ⁎-representation. These characterizations provide answers to the question raised by M. Mori in 2021 on finding a sufficient and necessary condition for a ⁎-algebra to have a unique C⁎-norm. For discrete groups, we give a characterization for a group algebra to have a unique C⁎-norm in terms of the group's unitary representations, providing an answer to the question raised by Alekseev in 2019 on characterizing C⁎-unique discrete groups.

On ϕ-P-flat modules and ϕ-von Neumann regular rings

Let [Formula: see text] be a commutative ring with a nonzero identity and [Formula: see text] an [Formula: see text]-module. Set [Formula: see text], if [Formula: see text]-[Formula: see text] then [Formula: see text] is called a [Formula: see text]-torsion module. An [Formula: see text]-module [Formula: see text] is said to be [Formula: see text]-flat, if [Formula: see text] is an exact [Formula: see text]-sequence, for any exact sequence of [Formula: see text]-modules [Formula: see text], where [Formula: see text] is [Formula: see text]-torsion. In this paper, we study some new properties of [Formula: see text]-flat modules. Then we introduce and study the class of [Formula: see text]-[Formula: see text]-flat modules which is a generalization of [Formula: see text]-flat modules and [Formula: see text]-flat modules. Finally, we give some new characterizations of the [Formula: see text]-von Neumann regular ring and its transfer to various contexts of constructions such as the amalgamation of rings along an ideal and trivial ring extension.

Bernstein-Sato theory for singular rings in positive characteristic

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case.

$S$-cotorsion modules and dimensions

Let $R$ be a ring, $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be $u$-$S$-flat ($u$- always abbreviates uniformly) if ${\rm Tor}^R_1 (M, N)$ is $u$-$S$-torsion $R$-module for all $R$-modules $N$. In this paper, we introduce and study the concept of $S$-cotorsion module which is in some way a generalization of the notion of cotorsion module. An $R$-module $M$ is said to be $S$-cotorsion if ${\rm Ext}^1_R(F,M)=0$ for any $u$-$S$-flat module $F$. This new class of modules will be used to characterize $u$-$S$-von Neumann regular rings. Hence, we introduce the $S$-cotorsion dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. As applications, we give a new upper bound on the global dimension of rings.

Surjectivity of some local cohomology map and the second vanishing theorem

The second vanishing theorem has a long history in the theory of local cohomology modules, which connects the vanishing of a complete regular local ring with a topological property of the punctured spectrum of the ring under some conditions. However, the case of complete ramified regular local rings is unresolved. In this paper, we give a partial answer to the second vanishing theorem in the ramified case. Our proof is inspired by the theory of surjective elements in the theory of local cohomology.

A Detailed Study of Mathematical Rings in q-Rung Orthopair Fuzzy Framework

Symmetry-related problems can be addressed by means of group theory, and ring theory can be seen as an extension of additive group theory. Ring theory, a significant topic in abstract algebra, is currently active in a diverse range of study domains across the disciplines of mathematics, theoretical physics and coding theory. The study of ideals is vital to the theory of rings in a wide range of ways. The uncertainties present in the information are addressed well by the q-rung orthopair fuzzy set (q-ROFS). Considering the significance of ring theory and the q-ROFS, this article defines q-rung orthopair fuzzy ideals (q-ROFIs) in conventional rings and investigates its various algebraic features. We introduce the notion of q-rung orthopair fuzzy cosets (q-ROFCs) of a q-ROFI and demonstrate that, under certain binary operations, the collection of all q-ROFCs of a q-ROFI forms a ring. In addition, we provide a q-rung orthopair analog of the fundamental theorem of ring homomorphism. Furthermore, we present the notion of q-rung orthopair fuzzy semi-prime ideals (q-ROFSPIs) and provide a comprehensive explanation of their many algebraic properties. Finally, regular rings were characterized using q-ROFIs.

TLidar-based crown shape indicates tree ring pattern in Norway spruce (Picea abies (L.) H. Karst) trees across competition gradients. A modeling and methodological approach

Tree crowns and growth rings are physiologically and functionally connected through supporting and resource sharing. Management interventions may strongly influence tree growth by altering this linkage. However, conventional approaches have limited ability to characterize crown shape precisely, thus hindering our understanding of the relationship between crown shape and tree ring patterns. We, thus, aimed to test three hypotheses: (HI) Crown shape (regularity vs. irregularity) and ring patterns (regularity or irregularity) are significantly correlated and (HII) vary across density gradients; if so, (HIII) internal ring patterns could be predicted from external crown shape metrics. We, therefore, employed terrestrial laser scan-based crown shape and coring-based tree ring width metrics for Norway spruce (Picea abies (L.) H. Karst.) trees covering a range of density gradients to assess temporal changes of crown shape and tree ring patterns. We found a significant and positive influence of crown shape quantifying metrics on ring patterns, indicating crown regularity or irregularity strongly reflects tree ring regularity or irregularity (p < 0.05). Crown shape and ring patterns always showed comparable patterns across density gradients (e.g., trees from lower-density stands produced transgressive crown and ring growth) and significantly varied across competition level. Trees grown in lower-density stands are more likely to produce upper-reaching crowns (maximum crown radius expansion shifted to the mid- to upper-crown) than trees grown in competitive conditions, which result in lower-reaching crowns (maximum crown radius shifted to the crown base) with reduced crown shape and ring pattern parameters. Crown irregularities increased as density decreased through competition reduction, resulting in more regular ring patterns (stable growth). Since both crown shape and ring patterns are simultaneously impacted by stand density or competition, the relationship between crown shape and ring patterns is competition-neutral. When viewed separately, both patterns had a strong relationship with the competition index. Finally, our comparative model predictions showed that approaches ranging from simple linear models to complex machine learning techniques (e.g., random forest, neural network, support vector machine, etc.) were effective in predicting ring patterns using external TLidar-crown shape, indicating a potential method to evaluate the crown shape and ring pattern link. The relationship between the crown and growth ring and their synchronous patterns across competition gradients suggests that internal growth can be assessed from the external appearances of trees and recommends further consideration in forest modeling.

On the K-theory of regular coconnective rings

We show that for a coconnective ring spectrum satisfying regularity and flatness assumptions, its algebraic K-theory agrees with that of its pi _0. We prove this as a consequence of a more general devissage result for stable infinity categories. Applications of our result include giving general conditions under which K-theory preserves pushouts, generalizations of mathbb {A}^n-invariance of K-theory, and an understanding of the K-theory of categories of unipotent local systems.

The lattice coordinatized by a ring

A coordinatization functor L ( − , 1 ) : Ring → Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functor are 1) that it extends the functor R ↦ L ( R ) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective ( − ) op endofunctors on Ring and Latt ; and 3) it admits localization at a left R -module. The complemented elements of L ( R , 1 ) form a partially ordered set S ( R ) isomorphic to the space of direct summands of R R . The right nonsingular rings for which the embedding of S ( R ) into the localization L ( R , 1 ) Q at the right maximal ring of quotients Q R is an isomorphism are characterized by a property that every finite matrix subgroup φ ( R R ) of the left R -module R R is essential in an element of S ( R ) . In that case, the space S ( R ) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients.

Malcolmson semigroups

Inspired by the construction of the Cuntz semigroup for a C⁎-algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic Grothendieck group in general and be isomorphic for von Neumann regular rings. For unital C⁎-algebras, it is shown that the matrix Malcolmson semigroup has a natural surjective order-preserving homomorphism to the Cuntz semigroup, every dimension function is a Sylvester matrix rank function, and there exist Sylvester matrix rank functions which are not dimension functions.

On accumulation points of F-pure thresholds on regular local rings

Blickle, Mustaţă and Smith proposed two conjectures on the limits of F-pure thresholds. One conjecture asks whether or not the limit of a sequence of F-pure thresholds of principal ideals on regular local rings of fixed dimension can be written as an F-pure thresshold in lower dimension. Another conjecture predicts that any F-pure threshold of a formal power series can be written as the F-pure threshold of a polynomial. In this paper, we prove that the first conjecture has a counterexample but a weaker statement still holds. We also give a partial affirmative answer to the second conjecture.