Geodesic
Closure operators in modules and their characterizations
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2020), pp. 31-62.

Voir la notice de l'article provenant de la source Math-Net.Ru

This work is dedicated to the investigation of closure operators of a module category $R$-Mod. The principal types of closure operators of $R$-Mod are studied and their characterizations are indicated, using dense or (and) closed submodules. The method of description of the closure operators consists in the elucidation of properties of functions which separate in every module, the set of dense submodules and the set of closed submodules. The main properties of the closure operators of $R$-Mod are studied: weakly heredity – idempotency, maximality – minimality, heredity – coheredity, as well as diverse combinations of them. Altogether, 16 types of the closure operators are described, among which 7 types possess double characterizations (by dense submodules and by closed ones). 
@article{BASM_2020_1_a2,
     author = {A. I. Kashu},
     title = {Closure operators in modules and their characterizations},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {31--62},
     publisher = {mathdoc},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2020_1_a2/}
}
TY  - JOUR
AU  - A. I. Kashu
TI  - Closure operators in modules and their characterizations
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2020
SP  - 31
EP  - 62
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BASM_2020_1_a2/
LA  - en
ID  - BASM_2020_1_a2
ER  - 
%0 Journal Article
%A A. I. Kashu
%T Closure operators in modules and their characterizations
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2020
%P 31-62
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BASM_2020_1_a2/
%G en
%F BASM_2020_1_a2
A. I. Kashu. Closure operators in modules and their characterizations. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2020), pp. 31-62. http://geodesic.mathdoc.fr/item/BASM_2020_1_a2/

[1] Bican L., Kepka T., Nemec P., Rings, modules and preradicals, Marcel Dekker, New York, 1982 | MR | Zbl

[2] Dikranjan D., Giuli E., “Closure operators I”, Topology and its Applications, 27 (1987), 129–143 | DOI | MR | Zbl

[3] Dikranjan D., Giuli E., Tholen W., “Closure operators II”, Proc. Intern. Conf. on Categorical Topology (Prague, 1988), World Scientific Publ., Singapore, 1989 | MR

[4] Dikranjan D., Tholen W., Categorical structure of closure operators, Kluwer Academic Publishers, 1995 | MR | Zbl

[5] Kashu A. I., “Closure operators in the categories of modules. Part I; II; III; IV”, Algebra and Discrete Math., 15:2 (2013), 213–228 ; Algebra and Discrete Math., 16:1 (2013), 81–95 ; Bul. Acad. Ştiinţe Repub. Moldova, Mat., 2014, no. 1(74), 90–100 ; Bul. Acad. Ştiinţe Repub. Moldova, Mat., 2014, no. 3(76), 13–22 | MR | Zbl | MR | Zbl | Zbl | Zbl

[6] Kashu A. I., “Preradicals, closure operators in $R$-Mod and connection between them”, Algebra and Discrete Math., 18:1 (2014), 86–96 | MR | Zbl

[7] Kashu A. I., “Pretorsions in modules and associated closure operators”, Bul. Acad. Ştiinţe Repub. Moldova, Mat., 2017, no. 2(84), 24–41 | MR | Zbl