Class Of Rings Research Articles

Non-Abelian Extensions of Groupoids and Their Groupoid Rings

We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids N→E→G gives rise to a groupoid crossed product of G by the groupoid ring of N which recovers the groupoid ring of E up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products.

Content and Q-sequences in mixed characteristic local rings

It is proved that a system of parameters is always a Q-sequence (in the sense of Hochster and Huneke) for several classes of mixed characteristic rings: rings in which the characteristic of the residue field is a nilpotent element, and mixed characteristic analogues of Stanley-Reisner rings, semigroup rings, and toric face rings.

Pairs of rings with the same ideal and Kronecker function rings

In this paper, we study the classical Kronecker function rings in a pair of rings R ⊆ T sharing the same ideal M, assumed to be maximal in T and contained in the Jacobson radical of R or that of T. We apply this to the D + M construction and the b-operations on D and R.

Units of twisted group rings and their correlations to classical group rings

This paper is centred around the classical problem of extracting properties of a finite group G from the ring isomorphism class of its integral group ring ZG. This problem is considered via describing the unit group U(ZG) generically for a finite group. Since the ‘90s’ several well known generic constructions of units are known to generate a subgroup of finite index in U(ZG) if QG does not have so-called exceptional simple epimorphic images, e.g. M2(Q). However it remained a major open problem to find a generic construction under the presence of the latter type of simple images. In this article we obtain such generic construction of units. Moreover, this new construction also exhibits new properties, such as providing generically free subgroups of large rank. As an application we answer positively for several classes of groups recent conjectures on the rank and the periodic elements of the abelianisation U(ZG)ab. To obtain all this, we investigate the group ring RΓ of an extension Γ of some normal subgroup N by a group G, over a domain R. More precisely, we obtain a direct sum decomposition of the (twisted) group algebra of Γ over the fraction field F of R in terms of various twisted group rings of G over finite extensions of F. Furthermore, concrete information on the kernel and cokernel of the associated projections is obtained. Along the way we also launch the investigations of the unit group of twisted group rings and of U(RΓ) via twisted group rings.

On some extensions of \pi -regular rings

Some variations of \pi -regular and nil clean rings were recently introduced in the works of the first author: “A generalization of \pi -regular rings, Turkish J. Math. 43 (2), 702–711 (2019)”, “A symmetrization in \pi -regular rings, Trans. A. Razmadze Math. Inst. 174 (3), 271–275 (2020)”, “A symmetric generalization of \pi -regular rings, Ric. Mat. 73 (1), 179–190 (2024)”. In this paper, we examine the structure and relationships between these classes of rings. Specifically, we prove that (m, n)-regularly nil clean rings are left-right symmetric and also show that the inclusions (D- regularly nil clean) (regularly nil clean) ((m, n)-regularly nil clean) hold, as well as we answer Questions 1, 2 and 3 posed in the third of the above-listed works. Moreover, we also consider other similar questions concerning the symmetric properties of certain classes of rings. For example, it is proven that centrally Utumi rings are always strongly \pi -regular.

Structurally Diverse Limonoids from Trichilia connaroides and Their Antitumor Activities

Comprehensive SummaryTwelve new limonoids (1—12), named trichilitins A—L, were isolated from the leaves and twigs of Trichilia connaroides, together with ten known compounds (13—22). The structures were elucidated by extensive spectroscopic investigations, X‐ray diffraction analyses, and ECD calculations. Compound 1, which belongs to a unique class of ring B‐seco limonoid, has been identified as 6/7/6/5 tetracyclic due to a key Baeyer‐Villiger oxidation. Compounds 2—7 were identified as ring intact limonoids, while compounds 8—10 were established as ring D‐seco ones, and 11 and 12 were determined to be rearranged ones. All of the compounds were tested for cytotoxicity against three human tumor cell lines (HCT‐116, NCl‐H1975, and SH‐SY5Y). Compounds 6, 7, 13, 14, and 19 exhibited significant cytotoxic effects, especially 7 exhibited significant cytotoxic effects against HCT‐116 with an IC50 value of 0.035 μmol·L–1 and was more active than the positive control, doxorubicin with an IC50 value of 0.20 μmol·L–1. Compound 7 effectively induced apoptosis of HCT‐116, which was associated with S‐phase cell cycle arrest. Furthermore, the Western blot analysis showed that compound 7 could induce cell cycle arrest by promoting the expression levels of p53 and p21.

Frobenius Local Rings of Order p4m

Suppose R is a finite commutative local ring, then it is known that R has four positive integers p,n,m,k called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field Fpm and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order p4m. Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants p,n,m,k. Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings.

Nonunion of Vascularized Fibula With Shortening and Multiplanar Deformity in a 6 Years Old Managed Simultaneously With Hexapod Circular External Fixator

The hexapod circular external fixator is considered a major upgrade to the classic Ilizarov ring fixators, which made it easier to manage complex deformities. Surgeons now, using it, can manage deformities in all three planes simultaneously without the need for frame modifications. We present a case report that shows the versatile nature of the hexapod circular fixator in a 6-year-old female patient who presented with multiple problems seen in a single bone, namely, two types nonunion at two different sites of a vascularized fibula graft of the leg (hypertrophic nonunion at the proximal site and atrophic nonunion at the distal site) with varus, procurvatum, and internal rotation deformities as well as 3-cm shortening. She underwent vascularized fibular graft surgery to treat osteomyelitis of the tibia 2 years prior. A hexapod circular fixator was applied which would statically stabilize the lower nonunion site (augmented with bone marrow injection at that site) and corrected the deformities and shortening through distraction at the proximal hypertrophic nonunion site in one surgical procedure. The case demonstrates the versatility of the hexapod circular fixator in treating complex orthopaedic situations, maintaining a minimally invasive approach while allowing full weight-bearing.

A sub-functor for Ext and Cohen–Macaulay associated graded modules with bounded multiplicity-II

Abstract Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and then the notion of a $T$ -split sequence was introduced in the part-1 of this paper for the $\mathfrak{m}$ -adic filtration with the help of the numerical function $e^T_A$ . In this article, we explore the relation between Auslander–Reiten (AR)-sequences and $T$ -split sequences. For a Gorenstein ring $(A,\mathfrak{m})$ , we define a Hom-finite Krull–Remak–Schmidt category $\mathcal{D}_A$ as a quotient of the stable category $\underline{\mathrm{CM}}(A)$ . This category preserves isomorphism, that is, $M\cong N$ in $\mathcal{D}_A$ if and only if $M\cong N$ in $\underline{\mathrm{CM}}(A)$ .This article has two objectives: first objective is to extend the notion of $T$ -split sequences, and second objective is to explore the function $e^T_A$ and $T$ -split sequences. When $(A,\mathfrak{m})$ is an analytically unramified Cohen–Macaulay local ring and $I$ is an $\mathfrak{m}$ -primary ideal, then we extend the techniques in part-1 of this paper to the integral closure filtration with respect to $I$ and prove a version of Brauer–Thrall-II for a class of such rings.

Local log-regular rings vs. toric rings

Local log-regular rings are a certain class of Cohen-Macaulay local rings that are treated in logarithmic geometry. Our paper aims to provide purely commutative ring theoretic proof of several ring-theoretic properties of local log-regular rings such as an explicit description of a canonical module, and the finite generation of the divisor class group.

Modules with reduced endomorphism rings

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including [Formula: see text]-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings [Formula: see text] for which every faithful multiplication [Formula: see text]-module is endo-reduced is precisely that of reduced rings; while the class of rings [Formula: see text] for which every multiplication [Formula: see text]-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

A new member of Nicholson's morphic folks

The main aim of the paper is to describe the structure of modules and corresponding rings satisfying the property (P), which says that M/im(α) is embeddable into ker⁡(α) for each endomorphism α and which generalizes the morphic property. In particular, it is proved that the class of rings with the property (P) is closed under taking products and summands and contains unit regular rings. We also explain connections between the virtually internal cancellation property and the property (P) and characterize the structure of particular classes of rings satisfying the property (P).

On the twisted group ring isomorphism problem for a class of groups

The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for certain associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special p-groups, and for all but five groups of order p5, where p≥5 is prime.

Rings whose clean elements are uniquely strongly clean

Abstract We define the class of CUSC rings, which are rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called USC rings, introduced by Chen–Wang–Zhou in J. Pure Appl. Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also generalize the so-called CUC rings, defined by Călugăreanu and Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. We establish that a ring is USC if and only if it is simultaneously CUSC and potent. Some other interesting relationships with CUC rings are obtained as well.

Validity and reliability of an immersive virtual reality adaptation of the 6-minute pegboard and ring test

Background and AimVirtual reality offers new clinical assessment and rehabilitation options that can complement or, in some cases, replace traditional methods. However, the applicability of using virtual reality tools for assessment of upper limb functional capacity has not been fully explored. We therefore developed an immersive virtual reality adaptation of the 6-Minute Pegboard and Ring Test (6PBRT-VR). The aim of the study was to test the validity and reliability of the 6PBRT-VR for the assessment of upper extremity functional capacity, and to assess the performance and feasibility of the proposed tool. MethodsThirty healthy young adults were included in the study. The participants performed the classical 6-Minute Pegboard and Ring Test first and then the 6PBRT-VR. The test-retest reliability of the 6PBRT-VR was assessed on intraclass correlation coefficient. Concurrent validity was assessed on the correlation between the 6PBRT-VR test-retest scores (number of rings moved) and the correlation between the scores from the classical 6-Minute Pegboard and Ring Test and the 6PBRT-VR. Convergent validity was assessed on correlations with handgrip strength and the total Quick Disabilities of the Arm, Shoulder, and Hand score. Cardiorespiratory responses were also measured (at baseline and after each test). Perceived arm fatigue was assessed on the Modified Borg Scale. ResultsThe 6PBRT-VR exhibited excellent test-retest reliability, with an intraclass correlation coefficient of 0.866 (95% confidence interval 0.737−0.934). Mean 6PBRT-VR score correlated strongly with the mean score of the classical 6-Minute Pegboard and Ring Test (r = 0.817, p < 0.001). A significant association was found between the 6PBRT-VR and the classical 6-Minute Pegboard and Ring Test in terms of variations in heart rate, systolic blood pressure, and Modified Borg Scale score (p < 0.001). Mean 6PBRT-VR score showed moderate correlations with right (r = 0.571, p = 0.001) and left handgrip strength (r = 0.550, p = 0.002). ConclusionThe 6PBRT-VR is a reliable and valid virtual tool for assessing upper-extremity functional capacity in young adults.

Product-complete tilting complexes and Cohen–Macaulay hearts

We show that the cotilting heart associated to a tilting complex T is a locally coherent and locally coperfect Grothendieck category (i.e., an Ind-completion of a small artinian abelian category) if and only if T is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring R . If \dim(R)&lt;\infty , we show that there is a derived duality \mathcal{D}^{\textup{b}}_{\textup{fg}}(R) \cong \mathcal{D}^{\textup{b}}(\mathcal{B})^{\textup{op}} between \mod R and a noetherian abelian category \mathcal{B} if and only if R is a homomorphic image of a Cohen–Macaulay ring. Along the way, we obtain new insights about t-structures in \mathcal{D}^{\textup{b}}_{\textup{fg}}(R) . In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional noetherian rings that admit a Gorenstein complex.

Rings in which every regular finitely generated ideal is $S$-finitely presented

In this work, we introduce and explore a class of rings where every regular finitely generated ideal is $S$-finitely presented, called a regular $S$-coherent ring. This concept represents a weaker version of the $S$-coherent ring property. It is shown that any $S$-coherent ring is inherently a regular $S$-coherent ring, and in the case of domains, the two properties are equivalent. We also investigate how this notion extends to different settings of commutative ring extensions, including direct products, trivial ring extensions, and the amalgamated duplication of a ring along an ideal. The obtained results yield new examples of regular $S$-coherent rings that are not $S$-coherent.

Flipped non-associative polynomial rings and the Cayley–Dickson construction

We introduce and study flipped non-associative polynomial rings. In particular, we show that all Cayley–Dickson algebras naturally appear as quotients of a certain type of such rings; this extends the classical construction of the complex numbers (and quaternions) as a quotient of a (skew) polynomial ring to the octonions, and beyond. We also extend some classical results on algebraic properties of Cayley–Dickson algebras by McCrimmon to a class of flipped non-associative polynomial rings.

Ring class fields and a result of Hasse

For squarefree d>1, let M denote the ring class field for the order Z[−3d] in F=Q(−3d). Hasse proved that 3 divides the class number of F if and only if there exists a cubic extension E of Q such that E and F have the same discriminant. Define the real cube roots v=(a+bd)1/3 and v′=(a−bd)1/3, where a+bd is the fundamental unit in Q(d). We prove that E can be taken as Q(v+v′) if and only if v∈M. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if v∈M, and we also show that the norm of the relative discriminant of F(v)/F lies in {1,36} or {38,318} according as v∈M or v∉M. We then prove that v is always in the ring class field for the order Z[−27d] in F. Some of the results above are extended for subsets of Q(d) properly containing the fundamental units a+bd.

Some results on the total zero-divisor graph of a commutative ring

PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.