Filial rings on direct sums and direct products of torsion-free abelian groups
https://doi.org/10.22405/2226-8383-2021-22-1-200-212
Abstract
A ring whose additive group coincides with an abelian group G is called a ring on G. An abelian group G is called a TI-group if every associative ring on G is filial. If every (associative) ring on an abelian group G is an SI-ring (a hamiltonian ring), then G is called an SI-group (an SI_H-group). In this article, TI-groups, SI_H-groups and SI-groups are described in the following classes of abelian groups: almost completely decomposable groups, separable torsionfree groups and non-measurable vector groups. Moreover, a complete description of non-reduced TI-groups, SI_H-groups and SI-groups is given. This allows us to only consider reduced groups when studying TI-groups.
About the Authors
Ekaterina Igorevna Kompantseva
Moscow Pedagogical State University; Financial University
under the Government of the Russian Federation
Russian Federation
Russian Federation
Nguyen Thi Quynh Trang
Vietnam education cooperation joint stock company
Viet Nam
Viet Nam
Varvara Aramovna Gazaryan
Moscow State University named after M.V. Lomonosov;
Financial University under the Government of the Russian Federation
Russian Federation
Russian Federation
References
1. Beaumont, R. A. 1948, “Rings with additive groups which is the direct sum of cyclic groups”, Duke Math. J., vol. 15, no. 2, pp. 367-369.
2. Feigelstock, S. 1997, “Additive groups of rings whose subrings are ideals”, Bull. Austral. Math. Soc., vol. 55, pp. 477-481.
3. Redei, L. 1952, “Vollidealringe im weiteren Sinn. I”, Acta Math. Acad. Sci. Hungar., vol. 3, pp. 243-268.
4. Andriyanov, V. I. 1967, “Periodic Hamiltonian rings”, Math. USSR-Sb., vol. 3, no. 2, pp. 225-242.
5. Kruse, R. L. 1968, “Rings in which all subrings are ideals”, Canad. J. Math., vol. 20, pp. 862-871.
6. Ehrlich, G. 1983-1984. “Filial rings”, Portugal. Math., vol. 42, pp. 185-194.
7. Sands, A. D. 1988, “On ideals in over-rings”, Publ. Math. Debrecen., vol. 35, pp. 273-279.
8. Andruszkiewicz, R. & Puczylowski, E. 1988, “On filial rings”, Portugal. Math., vol. 45, no. 2, pp. 139-149.
9. Filipowicz, M. & Puczylowski, E. R. 2004, “Left filial rings”, Algebra Colloq., vol. 11, pp. 335-344.
10. Filipowicz, M. & Puczylowski, E. R. 2005, “On filial and left filial rings”, Publ. Math. Debrecen, vol. 66, pp. 257-267.
11. Andruszkiewicz, R. & Pryszczepko, K. 2019, “On fully filial torsion rings” Bull. Korean Math. Soc., vol. 56, no. 1, pp. 23-29.
12. Baer, R. 1957, “Meta ideals”, Report conf. linear algebras. June. 1956., Publ. National Acad. Sci. nat. Res. Counil, no.502, pp. 33-52. 13. Andruszkiewicz, R. & Woronowicz, M. 2014, “On TI-groups”, Recent Results in Pure and Applied Math. Podlasie., pp. 33-41.
13. Andruszkiewicz, R. & Woronowicz, M. 2015, “On SI-groups”,Bull. of the Australian Math. Soc., vol. 91, no. 1, pp. 92-103.
14. Kompantseva, E. I. & Nguyen, T. Q. T. 2019, “Algebraically compact abelian TI-groups”, Chebyshevskii sbornik, vol. 20, no. 1, pp. 202-211.
15. Andruszkiewicz, R. & Woronowicz, M. 2017, “On additive groups of associative and commutative rings”, J. Quaest. Math., vol. 40, no. 4, pp. 527–537.
16. Fuchs, L. 2015, “Abelian groups”, Switz.: Springer International Publishing.
17. Mader, A. 2000, “Almost Completely Decomposable Abelian Groups”, Amsterdam: Gordon and Breach (Algebra Logic Appl. Ser., vol. 13).
18. Arnold, D. M. & Mader, A. & Mutzbauer, O. & Solak, E. 2012, “Almost completely decomposable groups and unbounded representation type”, J. of Algebra, vol. 349(1), pp. 50-62.
19. Solak, E. 2016, “Classification of a class of torsion-free abelian groups”, Math. Journal of the Univ. of Padova, vol. 135, pp. 111–131.
20. Kompantseva, E. I. 2009, “Rings on almost completely decomposable Abelian groups”, J. of Mathematical Sciences, vol. 163, no. 6, pp. 688-693.
21. Kompantseva, E. I. & Fomin, A. A. 2015, “Absolute ideals of almost completely decomposable abelian groups”, Chebyshevskii sbornik, vol. 16, no. 4, pp. 200–211.
22. Chekhlov, A. R. 2009, “On abelian groups, in which all subgroups are ideals”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., no. 3(7), pp. 64-67.
23. Baer, R. 1937, “Abelian groups without elements of finite order”, Duke Math. J., vol. 3, pp. 68-122.
24. Sasiada, E. 1959, “On the isomorphism of decompositions of torsion-free abelian groups into complete direct sums of groups of rank one”, Bull. Acad. Polon. Sci., vol. 7, pp. 145-149.
25. Mishina, A. P. 1962, “Separability of fully direct sums of abelian groups of rank 1”, Math. Sbornik, vol. 57, pp. 375-383.
Review
For citations:
Kompantseva E.I., Trang N.T., Gazaryan V.A. Filial rings on direct sums and direct products of torsion-free abelian groups. Chebyshevskii Sbornik. 2021;22(1):200-212. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-1-200-212
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