Preradicals and closure operators in modules: comparative analysis and relations
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2024-02-04 18:23
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KASHU, A., JARDAN, Jardan. Preradicals and closure operators in modules: comparative analysis and relations. In: Conference on Applied and Industrial Mathematics: CAIM 2022, Ed. 29, 25-27 august 2022, Chişinău. Chișinău, Republica Moldova: Casa Editorial-Poligrafică „Bons Offices”, 2022, Ediţia a 29, pp. 139-141. ISBN 978-9975-81-074-6.
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Conference on Applied and Industrial Mathematics
Ediţia a 29, 2022
Conferința "Conference on Applied and Industrial Mathematics"
29, Chişinău, Moldova, 25-27 august 2022

Preradicals and closure operators in modules: comparative analysis and relations


Pag. 139-141

Kashu A.1, Jardan Jardan12
 
1 Vladimir Andrunachievici Institute of Mathematics and Computer Science,
2 Technical University of Moldova
 
Disponibil în IBN: 21 decembrie 2022


Rezumat

The theory of radicals in modules is based by the notion of preradical (as subfunctor of identical functor) [1]. The other important notion of the modern algebra is the closure operator (as a function C which by every pair of modules N ⊆ M defines a submodule CM(N) ⊆ M, C being compatible by the morphisms of R-Mod) [2]. The purpose of this communication consists in the elucidation of the relations between these fundamental notions and the comparison of results of those respective theories. The closure operators in some sense are the generalization of preradicals, since the class PR(R) can be inserted in CO(R) (by two methods). This important fact determines a close connection between the results of the respective domains. In particular, there exists some correspondences between the main types of preradicals of R-Mod and respective types of closure operators. Also there exists a remarkable connection between the operations in the big lattices PR(R) and CO(R). These facts show the parallelism and similarity of two theories. However, it is obvious that CO(R) is essentially ,,larger” than PR(R) (closure operators are the functions of two variables). Therefore in the study of closure operators we must apply both the classical methods of radical theory, adding the constructions, notions and applications, related by the specificity of closure operators. In continuation we formulate some typical results of this domain. Theorem 1. There exists a monotone bijection between: a) the maximal closure operators of R-Mod and the preradicals of R-Mod: Max[CO(R)] ∼= PR(R); b) the minimal closure operators of R-Mod and the preradicals of RMod: Min[CO(R)] ∼= PR(R); b) the equivalence classes of CO(R), determined by the relation ,,∼” (C ∼ D def ⇐⇒ rC = rD), and the preradicals of R-Mod: CO(R)/∼ ∼= PR(R). Theorem 2. There exists a monotone bijection between: a) the idempotent preradicals of R-Mod and the closure operators which are maximal and weakly hereditary; b) the radicals of R-Mod and the closure operators which are maximal and idempotent; c) the pretorsions (torsions) of R-Mod and the closure operators which are minimal and hereditary (maximal, idempotent and hereditary). Theorem 3. There exists a monotone bijection between the cohereditary closure operators of R-Mod and the ideals of R.