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Weakly injective modules and weakly projective modules

https://doi.org/10.26907/0021-3446-2025-12-3-26

Abstract

Weakly injective and weakly projective modules, which are extensions of the classes of quasi-injective modules and quasi-projective modules respectively, are studied. Descriptions are obtained for weakly injective and weakly projective Abelian groups, as well as for some classes of Abelian groups closely related to them.

About the Authors

A. N. Abyzov
Kazan State University
Russian Federation

Adel Nailevich Abyzov

18 Kremlyovskaya str., Kazan, 420008



T. D. Bui
Kazan State University; Can Tho University of Technology
Russian Federation

Tien Dat Bui

18 Kremlyovskaya str., Kazan, 420008,

256 Nguyen Van Cu str., Can Tho City



References

1. Johnson R.E., Wong E.T. Quasi-injective modules and irreducible rings, J. London Math. Soc. 36 (1), 260–268 (1961).

2. Jain S.K., Singh S. On pseudo-injective modules and self pseudo injective rings, J. Math. Sci. 2 (1), 23–31 (1967).

3. Dickson S.E., Fuller K.R. Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math. 31 (3), 655–658 (1969).

4. Er N., Singh S., Srivastava A. Rings and modules which are stable under automorphisms of their injective hulls, J. Algebra 379, 223–229 (2013).

5. Santa-Clara C. Extending modules with injective or semisimple summands, J. Pure Appl. Algebra 127 (2), 193–203 (1998).

6. Abyzov A.N., Quynh T.C., Ha N.T.T, Yildirim T. Modules close to the automorphism-invariant and coinvariant, J. Algebra Appl. 18 (12), 1950235 (2019).

7. Tuganbaev A.A. Automorphism-invariant semi-Artinian modules. J. Algebra Appl. 16 (2), 1750029 (2017).

8. Tuganbaev A.A. Semidistributive modules and rings (Kluwer Acad. Publ. Dordrecht, Boston, London, 1998).

9. Абызов А.Н., Буй Т.Д. Существенно квазиинъективные модули и их прямые суммы, Изв. вузов. Матем. (7), 9–23 (2024).

10. Туганбаев А.А. Характеризации колец, использующие малоинъективные и малопроективные модули, Вестн. МГУ. Сер. Матем., мех. (3), 48–51 (1979).

11. Singh S., Srivastava A.K. Dual automorphism-invariant modules, J. Algebra 371, 262–275 (2012).

12. Abyzov A.N., Quynh T.C. Lifting of automorphisms of factor modules, Commun. Algebra 46 (11), 5073–5082 (2018).

13. Мишина А.П. Об автоморфизмах и эндоморфизмах абелевых групп, Вестн. Моск. ун-та. Сер. 1. Матем. Механ. (4), 39–43 (1962).

14. Мишина А.П. Об автоморфизмах и эндоморфизмах абелевых групп. II, Вестн. Моск. ун-та. Сер. 1. Матем. Механ. (1), 62–66 (1972).

15. Tuganbaev A.A. Arithmetical Rings and Endomorphisms (Walter de Gruyter, Berlin-Boston, 2019).

16. Tuganbaev A.A. Automorphism-extendable and endomorphism-extendable modules, J. Math. Sci. 245 (2), 234–284 (2020).

17. Кильп М.А. Квазиинъективные абелевы группы. Вестн. Моск. ун-та. Сер. 1. Матем. Механ. (3), 3–4 (1967).

18. Fuchs L., Rangaswamy K.M. Quasi-projective abelian groups. Bull. Soc. Math. France 98, 5–8 (1970).

19. Fuchs L. Infinite Abelian Groups. I, Pure Appl. Math. 36 (Acad. Press, 1970).

20. Burgess W.D., Raphael R. On modules with the absolute direct summand property, 137–148 (World Sci. Publ., River Edge, 1993).

21. Ko¸san T.M., Zemliˇcka J. ˇ ADS Abelian groups, J. Algebra Appl., 2450182 (2023).

22. Trang D.T., Ko¸san T.M., Tasdemir O., Quynh T.C. On modules and rings having large absolute direct summands, Comm. Algebra 51 (12), 4949–4961 (2023).

23. Туганбаев А.А. Теория колец. Арифметические модули и кольца (М., МЦНМО, 2009).

24. Wisbauer R. Foundations of module and ring theory (Gordon and Sci. Publ., 1991).

25. Каш Ф. Модули и кольца (Мир, М., 1981).

26. Dung N.V., Huynh D.V., Smith P.F., Wisbauer R. Extending modules (SRC Press, 1994).

27. Quynh T.C., Ko¸san M.T. On ADS modules and rings, Commun. Algebra 42 (8), 3541–3551 (2014).

28. Fuchs L. Abelian Groups (Springer Monographs Mathematics, 2015).

29. Mohamed S.H., M¨uller B.J. Continuous and discrete modules 147 (Cambridge Univ. Press, 1990).

30. Абызов А.Н., Буй Т.Д. Мало квазипроективные модули, Матем. Теорет. комп. науки 2 (3), 4–28 (2024).

31. Tuganbaev A.A. Automorphism-liftable modules, J. Algebra Appl. 21 (6), 2250106 (2022).

32. Clark J., Lomp C., Vanaja N., Wisbauer R. Lifting modules: Supplements and projectivity in module theory (Birkhauser Verlag, Basel, 2006).

33. Абызов А.Н., Куинь Ч.К., Туганбаев А.А. Модули, инвариантные относительно автоморфизмов и идемпотентных эндоморфизмов своих оболочек и накрытий, Итоги науки и техн. Серия Совр. матем. и ее прил. Темат. обз. 159, 3–45 (2019).

34. Feigelstock S., Raphael R. A Problem on relative projectivity for Abelian groups. Canad. Math. Bull. 29 (1), 114–122 (1986).


Review

For citations:


Abyzov A.N., Bui T.D. Weakly injective modules and weakly projective modules. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2025;(12):3-26. (In Russ.) https://doi.org/10.26907/0021-3446-2025-12-3-26

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ISSN 0021-3446 (Print)
ISSN 2076-4626 (Online)