A Note on Abelian Groups Supporting Unital Rings

Mateusz Woronowicz

doi:10.2478/amsil-2024-0016

.blurhash-client-img { display: none !important; }

A Note on Abelian Groups Supporting Unital Rings

Annales Mathematicae Silesianae

Volume 39 (2025): Issue 1 (March 2025)

By: Mateusz Woronowicz

Open Access

|Jun 2024

Abstract

Non-nil abelian groups are classified on which every ring, different from the zero-ring, is unital. It is shown that the assumption on the associativity of the considered rings does not influence the obtained classification. A significant mistake made by other authors studied this topic is corrected.

References

R.R. Andruszkiewicz and M. Woronowicz, On associative ring multiplication on abelian mixed groups, Comm. Algebra 42 (2014), no. 9, 3760–3767.
Search in Google Scholar Back to article
R.R. Andruszkiewicz and M. Woronowicz, On SI-groups, Bull. Aust. Math. Soc. 91 (2015), no. 1, 92–103.
Search in Google Scholar Back to article
R.R. Andruszkiewicz and M. Woronowicz, The classification of torsion-free TI-groups, Algebra Colloq. 29 (2022), no. 4, 595–606.
Search in Google Scholar Back to article
R.A. Beaumont, Rings with additive group is the direct sum of cyclic groups, Duke Math. J. 15 (1948), 367–369.
Search in Google Scholar Back to article
S. Breaz and G. Călugăreanu, Additive groups of rings with identity. Avaliable at http://math.ubbcluj.ro/~calu/ident.pdf.
Search in Google Scholar Back to article
S. Feigelstock, Additive Groups of Rings, Vol. I, Res. Notes in Math., 83, Pitman (Advanced Publishing Program), Boston, MA, 1983.
Search in Google Scholar Back to article
S. Feigelstock, Additive Groups of Rings, Vol. II, Pitman Res. Notes Math. Ser., 169, Longman Scientific & Technical, London, 1988.
Search in Google Scholar Back to article
S. Feigelstock, Additive groups of rings whose subrings are ideals, Bull. Austral. Math. Soc. 55 (1997), no. 3, 477–481.
Search in Google Scholar Back to article
L. Fuchs, Infinite Abelian Groups, Vol. I, Pure Appl. Math., Vol. 36, Academic Press, New York-London, 1970.
Search in Google Scholar Back to article
L. Fuchs, Infinite Abelian Groups, Vol. II, Pure Appl. Math., Vol. 36-II, Academic Press, New York-London, 1973.
Search in Google Scholar Back to article
F. Hasani, F. Karimi, A. Najafizadeh, and Y. Sadeghi, On the square subgroups of decomposable torsion-free abelian groups of rank three, Adv. Pure Appl. Math. 7 (2016), no. 4, 259–265.
Search in Google Scholar Back to article
E.I. Kompantseva, Abelian dqt-groups and rings on them (in Russian), Fundam. Prikl. Mat. 18 (2013), no. 3, 53–67; translation in J. Math. Sci. (N.Y.) 206 (2015), no. 5, 494–504.
Search in Google Scholar Back to article
E.I. Kompantseva and T.Q.T. Nguyen, Algebraically compact abelian TI-groups (in Russian), Chebyshevskiĭ Sb. 20 (2019), no. 1, 204–213.
Search in Google Scholar Back to article
P.T.T. Thuy, Torsion Abelian RAI-groups (in Russian), Fundam. Prikl. Mat. 17 (2012), no. 8, 109–138; translation in J. Math. Sci. (N.Y.) 197 (2014), no. 5, 658–678.
Search in Google Scholar Back to article
M. Woronowicz, A note on the square subgroups of decomposable torsion-free abelian groups of rank three, Ann. Math. Sil. 32 (2018), no. 1, 319–331.
Search in Google Scholar Back to article
M. Woronowicz, A note on Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups in the context of additive groups of rings of finite rank, Bull. Belg. Math. Soc. Simon Stevin 27 (2020), no. 4, 509–519.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/amsil-2024-0016 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107

Journal RSS Feed

Language: English

Page range: 170 - 176

Submitted on: May 10, 2023

Accepted on: May 19, 2024

Published on: Jun 7, 2024

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

20K99,

16B99

Related subjects:

Mathematics,

General mathematics

© 2024 Mateusz Woronowicz, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 39 (2025): Issue 1 (March 2025)