Projective Modules Research Articles

Finitistic Dimensions of Artin Algebras with Two Simples and a Conjecture of Marczinzik

Let Λ be an Artin algebra with a unique non-injective indecomposable projective module. In this situation, Marczinzik conjectured that the dominant dimension of Λ agrees with its finitistic dimension. In this paper, we give a proof of a stronger statement. As a byproduct, we obtain excellent control over the finitistic dimensions of Artin algebras with two simples and positive dominant dimension, and also establish the Gorenstein symmetry conjecture for all algebras under consideration.

Relative copure modules and Foxby equivalence

Let R be a commutative ring and C a semidualizing R-module. For a non-negative integer n, we study a natural generalization of copure injective (flat) modules by introducing the concepts of n-C-copure projective (injective, flat) modules and the most important results about copure injective (flat) modules are generalized. Moreover, it is proven that the n-C-copure projective (injective, flat) module is a generalization of C-Gorenstein projective (injective, flat) module and they are the same when the C-Gorenstein global dimension of R is no more than n. At the end, these classes of modules are used to extend the Foxby equivalence.

Community partners' experiences of higher education service-learning in a community engagement module

PurposeThe purpose of this empirical research paper is to investigate the self-perceived role of the community partner of a higher education service-learning and community engagement module.Design/methodology/approachA qualitative approach was followed by distributing a questionnaire to the community partners of a community engagement module and coding the responses using ATLAS.ti. A total of 36 responses were received from community partners who work with students enrolled in a compulsory undergraduate community-based project module at the University of Pretoria's Faculty of Engineering, Built Environment and Information Technology.FindingsThe community partners share a common interest in the students' education. They are experts in their fields and can share their knowledge with the students and the university. Through these partnerships, long-term reciprocal relationships can develop. Community partners can become co-educators and partners in education. The pragmatist representations of community partners can be challenged when they understand their own stakes in service-learning or community engagement projects. This better aids higher education institutes in the management and evaluation of service-learning and community engagement pedagogies and curricula.Research limitations/implicationsTwo main limitations underlie this study. Firstly, this research is based on data from one community module at a single university. Although a large number of students are registered in the module, the study would be improved by conducting it at more than one university countrywide. Secondly, the study was performed during the first coronavirus disease 2019 (COVID-19) lockdown the country experienced. This was a completely unexpected event for which everyone was totally unprepared. Many of the community partners lacked the resources to receive or respond to an online questionnaire. The nature of the lockdown prevented the researchers from reaching these community partners for a face-to-face interview. The voice of these community partners is, therefore, silent.Practical implicationsThe community partners reiterated their need to be seen as equal partners in the module and appreciated being part of a group of non-profit enterprises working together with a university to pursue a set of common goals. However, their status as peers depends on their willingness and ability to contribute sufficiently to the structure and demands of the service-learning module. The community partners who were able and willing to orientate each group of students to their organisation's mission and objectives, and who executed their roles according to the course requirements, experienced the greatest success in terms of project effectiveness and efficiency, and also in terms of future benefits when students returned to volunteer or provide donations. Given time, these community partners grew into an equal partner with the university's stakeholders, where both their own needs and those of the students were met during the various service-learning projects.Social implicationsSince all respondents in this study are non-profit organisations, the financial assistance and free labour afforded to them by the students are of paramount importance. The community partners also understand the longer-term value implications of successful student projects, as some students return of their free will to volunteer their services when gainfully employed after graduation.Originality/valueCommunity engagement projects are rarely investigated from the community partner's point of view. This paper elicited their responses and examined them through the lens of Fraser's theory of social justice (Fraser, 2009).

The cancellation of projective modules of rank 2 with a trivial determinant

We study the cancellation property of projective modules of rank $2$ with a trivial determinant over Noetherian rings of dimension $\leq 4$. If $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ such that $6 \in k^{\times}$, then we prove that stably free $R$-modules of rank $2$ are free if and only if a Hermitian $K$-theory group $\tilde{V}_{SL} (R)$ is trivial.

Multimodal Infant Brain Segmentation by Fuzzy-Informed Deep Learning

Magnetic resonance imaging (MRI) is a prevailing method of modal infant brain tissue analysis that precisely segments brain tissue and is vitally important for diagnosis, remediation, and analysis of early brain development. To achieve such segmentation is challenging, particularly for the brain of a six-month-old, owing to several factors: poor image quality; isointense contrast between white and gray matter and the simple incomplete volume consequence of a tiny brain size; and discrepancies in brain tissues, illumination settings, and the vagarious region. This article addresses these challenges with a fuzzy-informed deep learning segmentation network that takes T1- and T2-weighted MRIs as inputs. First, a fuzzy logic layer encodes input to the fuzzy domain. Second, a volumetric fuzzy pooling (VFP) layer models the local fuzziness of the volumetric convolutional maps by applying fuzzification, accumulation, and defuzzification on the adjacency feature map neighborhoods. Third, the VFP layer is employed to design the fuzzy-enabled multiscale feature learning module to enable the extraction of brain features in different receptive fields. Finally, we redesign the Project & Excite module using the VPF layer to enable modeling uncertainty during feature recalibration, and a comprehensive training paradigm is used to learn the ideal parameters of every building block. Extensive experimental comparative studies substantiate the efficiency and accuracy of the proposed model in terms of different evaluation metrics to solve multimodal infant brain segmentation problems on the iSeg-2017 dataset.

Projectivity relative to closed (neat) submodules

An [Formula: see text]-module [Formula: see text] is called closed (neat) projective if, for every closed (neat) submodule [Formula: see text] of every [Formula: see text]-module [Formula: see text], every homomorphism from [Formula: see text] to [Formula: see text] lifts to [Formula: see text]. In this paper, we study closed (neat) projective modules. In particular, the structure of a ring over which every finitely generated (cyclic, injective) right [Formula: see text]-module is closed (neat) projective is studied. Furthermore, the relationship among the proper classes which are induced by closed submodules, neat submodules, pure submodules and [Formula: see text]-pure submodules are investigated.

On Almost Projective Modules

In this note, we investigate the relationship between almost projective modules and generalized projective modules. These concepts are useful for the study on the finite direct sum of lifting modules. It is proved that; if M is generalized N-projective for any modules M and N, then M is almost N-projective. We also show that if M is almost N-projective and N is lifting, then M is im-small N-projective. We also discuss the question of when the finite direct sum of lifting modules is again lifting.

McKay matrices for finite-dimensional Hopf algebras

AbstractFor a finite-dimensional Hopf algebra $\mathsf {A}$ , the McKay matrix $\mathsf {M}_{\mathsf {V}}$ of an $\mathsf {A}$ -module $\mathsf {V}$ encodes the relations for tensoring the simple $\mathsf {A}$ -modules with $\mathsf {V}$ . We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $\mathsf {M}_{\mathsf {V}}$ by relating them to characters. We show how the projective McKay matrix $\mathsf {Q}_{\mathsf {V}}$ obtained by tensoring the projective indecomposable modules of $\mathsf {A}$ with $\mathsf {V}$ is related to the McKay matrix of the dual module of $\mathsf {V}$ . We illustrate these results for the Drinfeld double $\mathsf {D}_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $\mathsf {M}_{\mathsf {V}}$ and $\mathsf {Q}_{\mathsf {V}}$ in terms of several kinds of Chebyshev polynomials. For the matrix $\mathsf {N}_{\mathsf {V}}$ that encodes the fusion rules for tensoring $\mathsf {V}$ with a basis of projective indecomposable $\mathsf {D}_n$ -modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

Equivariant vector bundles over quantum spheres

We quantize homogeneous vector bundles over an even complex sphere $\mathbb{S}^{2n}$ as one-sided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite $\mathbb{C}$-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to $\mathbb{S}^{2n}_q$ and prove their complete reducibility.

Mappings between the lattices of saturated submodules with respect to a prime ideal

Let $\mathfrak{S}_p(_RM)$ be the lattice of all saturated submodules of an $R$-module $M$ with respect to a prime ideal $p$ of a commutative ring $R$. We examine the properties of the mappings $\eta:\mathfrak{S}_p(_RR)\rightarrow \mathfrak{S}_p(_RM)$ defined by $\eta(I)=S_p(IM)$ and $\theta:\mathfrak{S}_p(_RM)\rightarrow \mathfrak{S}_p(_RR)$ defined by $\theta(N)=(N:M)$, in particular considering when these mappings are lattice homomorphisms. It is proved that if $M$ is a semisimple module or a projective module, then $\eta$ is a lattice homomorphism. Also, if $M$ is a faithful multiplication $R$-module, then $\eta$ is a lattice epimorphism. In particular, if $M$ is a finitely generated faithful multiplication $R$-module, then $\eta$ is a lattice isomorphism and its inverse is $\theta$. It is shown that if $M$ is a distributive module over a semisimple ring $R$, then the lattice $\mathfrak{S}_p(_RM)$ forms a Boolean algebra and $\eta$ is a Boolean algebra homomorphism.

Development of project and ICT-based learning media subject modules

The purpose of this study is to develop a valid project-based learning media module and ICT, to determine the practicality of learning using the project-based learning media module and practical information technology, to determine the effectiveness of learning using the project-based learning media module and information technology. This type of research is development research. The conclusions of this study are the project-based learning media module and valid ICT, project-based learning media and practical ICT modules, and learning using project-based learning m edia modules and effective I CT.

Research on Geological Information Collection Based on Data Mining

In order to improve the efficiency of collecting geological data and digitization, the geological data acquisition system based on WorldWind mobile terminal is developed, which integrates WorldWind map technology, network communication technology, multithread technology, and data storage technology. The realization methods of location module, data input module, solid projection module, and 3D display module are expounded. The research and development of this system realizes the information collection, browsing, and display of geological work, provides convenient data input mode for geological workers, and promotes the development of digitization, information, and modernization of geological and mineral work.

Generalized “stacked bases” theorem for modules over semiperfect rings

The history of generalized “stacked bases” theorem origins from the result of Hill and Megibben on abelian groups. We extend this theorem for modules over semiperfect rings and as a consequence we show that for a submodule H of a projective module G over a semiperfect ring, the following conditions are equivalent: there exists a decomposition into a direct sum of indecomposable modules Pi , such that G/H is a direct sum of a family of modules, isomorphic to factor modules of principal indecomposable modules.

Relative torsionfree modules with respect to a faithfully balanced bimodule

Let S and R be two associative rings, be a faithfully balanced (S, R)-bimodule. We introduce and investigate properties of ∞-torsionfree modules with respect to which are generalizations of n-torsionfree modules for any positive integer n and totally C-reflexive modules. And we give some characterizations of the class of the ∞- C-torsionfree modules over any ring R. Moreover, when C is a semidualizing (S, R)-bimodule, we study the relations between the classes of relative torsionfree modules and the torsionfree modules in terms of Foxby equivalence. Finally, as the ∞-C-torsionfree module which is in the left orthogonal class of C is exactly the C-Gorenstein projective module, we characterize the Auslander class with respect to a semidualizing module by virtue of the ∞-C-torsionfree modules over the commutative Cohen-Macaulay ring with a dualizing module, which is a module version of Theorem 4.6 in Holm, H., Jørgensen, P. (2006). [Semi-dualizing modules and related Gorenstein homologica dimensions. J.Pure Appl. Algebra 205(2):423–445].

Noncommutative Borsuk–Ulam-type conjectures revisited

Let H be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra A . It was recently conjectured that there does not exist an equivariant *-homomorphism from A (type-I case) or H (type-II case) to the equivariant noncommutative join C*-algebra A\circledast^\delta H . When A is the C*-algebra of functions on a sphere, and H is the C*-algebra of functions on \mathbb{Z}/2\mathbb{Z} acting antipodally on the sphere, then the conjecture of type I becomes the celebrated Borsuk–Ulam theorem. Taking advantage of recent work of Passer, we prove the conjecture of type I for compact quantum groups admitting a non-trivial torsion character. Next, we prove that, if the compact quantum group (H,\Delta) admits a representation whose K_1 -class is non-trivial and A admits a character, then a stronger version of the type-II conjecture holds: the finitely generated projective module associated with A\circledast^\delta H via this representation is not stably free. In particular, we apply this result to the q -deformations of compact connected semisimple Lie groups and to the reduced group C*-algebras of free groups on n>1 generators.

RSS equivalences over a class of Morita rings

For two bimodules NBA and MAB with M⊗AN=0=N⊗BM, the monomorphism category M(A,M,N,B) and its dual, the epimorphism E(A,M,N,B), are introduced and studied. By definition, M(A,M,N,B) is the subcategory of Δ-mod consisting of (X,Y,f,g) such that f:M⊗AX→Y is a monic B-map and g:N⊗BY→X is a monic A-map, where Δ=(ANMB) is a Morita ring. This monomorphism category is a resolving subcategory of Δ-mod if and only if MA and NB are projective modules; and if this is the case and if in addition HomB(N,B)A and HomA(M,A)B are projective modules, then it has enough relative injective objects, and it is a Frobenius category if and only if A and B are selfinjective. Under these conditions we prove that there is a unique cotilting left Δ-module T, up to the multiplicity of the indecomposable direct summands, such that M(A,M,N,B)=T⊥. When M and N are exchangeable bimodules, Ringel-Schmidmeier-Simson equivalence between M(A,M,N,B) and E(A,M,N,B) are given.

Finiteness conditions and relative derived categories

In this paper, we introduce a class of exact structures in terms of finiteness conditions of modules, which are called n-pure exact structures. We investigate the properties of n-pure derived categories of module categories using n-pure exact structures, and show that n-pure derived categories share many nice properties of classical derived categories. In particular, we show that bounded n-pure derived categories can be realized as certain homotopy categories. We also compare the classical bounded derived categories and the bounded n-pure derived categories, and show that the former can be described by n-pure projective modules.

Invariance of projective modules in $$\mathsf {Sup}$$ under self-duality

Invariance of projective modules in $$\mathsf {Sup}$$ under self-duality

$(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS

Let $R$ be a ring, $n$ be an non-negative integer and $d$ be a positive integer or $\infty$. A right $R$-module $M$ is called \emph{$(n,d)^*$-projective} if ${\rm Ext}^1_R(M, C)=0$ for every $n$-copresented right $R$-module $C$ of injective dimension $\leq d$; a ring $R$ is called \emph{right $(n,d)$-cocoherent} if every $n$-copresented right $R$-module $C$ with $id(C)\leq d$ is $(n+1)$-copresented; a ring $R$ is called \emph{right $(n,d)$-cosemihereditary} if whenever $0\rightarrow C\rightarrow E\rightarrow A\rightarrow 0$ is exact, where $C$ is $n$-copresented with $id(C)\leq d$, $E$ is finitely cogenerated injective, then $A$ is injective; a ring $R$ is called \emph{right $(n,d)$-$V$-ring} if every $n$-copresented right $R$-module $C$ with $id(C)\leq d$ is injective. Some characterizations of $(n,d)^*$-projective modules are given, right $(n,d)$-cocoherent rings, right $(n,d)$-cosemihereditary rings and right $(n,d)$-$V$-rings are characterized by $(n,d)^*$-projective right $R$-modules. $(n,d)^*$-projective dimensions of modules over right $(n,d)$-cocoherent rings are investigated.

Homological Algebra for Persistence Modules

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings and show how these theorems can be applied to persistence modules arising from filtered cell complexes. We also give a Gabriel-Popescu theorem for persistence modules. Finally, we examine categories enriched over persistence modules. We show that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself.