INTEGRATION OF DIGITAL AND PEDAGOGICAL TECHNOLOGIES IN THE PROCESS OF TEACHING STUDENTS OF SOCIO-ECONOMIC AND PEDAGOGICAL AREAS OF TRAINING

Тhis study examines how to integrate the benefits of role-playing, collaborative and interactive learning, and realistic case studies in a virtual environment. The article describes the experience of implementing an educational model based on the online virtual reality platform Second Life (SL), which provides students with the opportunity to participate in role-playing games, interactive learning, and social interaction in the learning process. The proposed model consists of three modules: the module for the preparation of information, joint modeling, a reflexive module that provides practical skills by participating in a 3D game. Students and teachers who participated in the testing of the model gave a positive assessment. The main goal of developing an educational model that's based on Second Life is to improve the educational process, using the advantages of role-playing games, collaborative and interactive learning, as well as realistic cases in a virtual environment. This can be achieved using three modules: an information preparation module, a joint simulation module for studying the situation, and a reflexive module. Unlike some virtual educational systems, this model is specifically designed to help students understand the causes of problems and how to solve them through data visualization and simple modeling, easy communication, flexible interaction, collaboration, and immersion in the SL environment. By integrating the three modules in the SL environment, the model will provide more opportunities for effective collaborative learning in various fields of knowledge. In the information preparation module, teachers and students use moving avatars to perform all the actions presented in the demo virtual room in SL with the support of the Semantic Wiki Template. This template provides the platform with sections of information on the studied subject. Using it, teachers can easily pass practical tasks to the student, while students can respond and present results without having any deep training in the field of information technology.

Acyclic complexes of finitely generated free modules over local rings

We consider the question of how minimal acyclic complexes of finitely generated free modules arise over a commutative local ring. A standard construction gives that every totally reflexive module yields such a complex. We show that for certain rings this construction is essentially the only method of obtaining such complexes. We also give examples of rings which admit minimal acyclic complexes of finitely generated free modules which cannot be obtained by means of this construction.

A Note on the Artinianness and Vanishing of Local Cohomology and Generalized Local Cohomology Modules

The first part of this paper is concerned with the Artinianness of certain local cohomology modules [Formula: see text] when M is a Matlis reflexive module over a commutative Noetherian complete local ring R and 𝔞 is an ideal of R. Also, we characterize the set of attached prime ideals of [Formula: see text], where n is the dimension of M. The second part is concerned with the vanishing of local cohomology and generalized local cohomology modules. In fact, when R is an arbitrary commutative Noetherian ring, M is an R-module and 𝔞 is an ideal of R, we obtain some lower and upper bounds for the cohomological dimension of M with respect to 𝔞.

An Uncountably Infinite Number of Indecomposable Totally Reflexive Modules

Several years ago, Huneke and Leuschke proved a theorem solving a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is considered over an arbitrary local ring. The main purpose of this paper is to prove that the existence of a certain prime ideal and a certain totally reflexive module implies the existence of an uncountably infinite number of isomorphism classes of indecomposable totally reflexive modules.

The natural topology of Matlis reflexive modules

For a complete local ring R with maximal ideal m, we define a linear topology on a Matlis reflexive R-module M which coincides with the m-adic topology on M in case M is finitely generated. We show that a Matlis reflexive module is complete in this topology.