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On some special constructions from theory of radicals in categories of modules


Author: Jardan Ion
Degree:doctor of
Speciality: 01.01.06 - Mathematical logics, algebra and theory of numbers
Year:2020
Scientific adviser: Alexei Caşu
doctor habilitat, professor, Institute of Mathematics and Computer Science
Institution: Institute of Mathematics and Computer Science

Status

The thesis was presented on the 18 September, 2020
Approved by NCAA on the 23 December, 2020

Abstract

Adobe PDF document – 0.26 Mb / in romanian

Thesis

CZU 512.548

Adobe PDF document 0.62 Mb / in romanian
106 pages

Keywords

ring, module, category, lattice, (pre)radical, (pre)torsion

Summary

The thesis „On some special constructions from theory of radicals in categories of modules” is presented by Jardan Ion to obtain the degree of PhD in mathematical sciences, speciality 111.03 – Mathematical logic, algebra and theory of numbers. It was elaborated at the Vladimir Andrunachievici Institute of Mathematics and Computer Science, Chiёsin˘au, 2020. Thesis structure: the work is written in Roumanian and consists of introduction, 5 chapters, general conclusions and recommendations, 133 bibliography items, 91 pages of main text. The obtained results were published in 9 scientific works. Keywords: ring, module, category, lattice, (pre)radical, (pre)torsion. Field of study of the thesis: theory of rings and modules, radicals in the categories of modules. The purpose and objectives: introduction of some new operations in the class of preradicals of a module category, elucidation of their properties and of preradicals obtained as a result of these constructions; investigation of some particular cases of new operations and of relations between the obtained preradicals; the study of connections between the new operations and some know notions and constructions of the radicals theory; the recearch of the behaviour of these operations in case of preradicals of special types. Scientific innovation and originality: all principal results of this work are new and original. They constitute a natural continuation of the previous investigations in this domain. Namely, the well known four operations in the class of preradicals are completed by four new operations. The properties of them are determined, the compatibility by the latticeal operations in the class of preradicals. The relations of these operations with some constructions of the theory of radicals are shown, the behaviour of them in the case of preradicals of special types is described. The obtained result which contributes to solve some important scientific problem consists of the introduction and the investigation of new four operations in the class of preradicals of a module category, which led to enrichment of instrumental reservs of work in this class, as well to complete the knowledge in theory of rings and modules. The theoretical significance and applicative value of the thesis: this work has a theoretical character and represents an important step in the development of the theory of radicals in modules. The importance of results consists in: the broadening of knowledges in domain by investigation of new operations with preradicals; the essencial completion of methods of investigation by new operations; the possibility of application of them in structural problems. The implementation of the scientific results: the obtained in the this work results can be used in the further development of the radical theory, also can serve as a support for the preparation of courses for university faculties.