A Note on Abelian Groups Supporting Unital Rings

Woronowicz, Mateusz

doi:10.2478/amsil-2024-0016

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

A Note on Abelian Groups Supporting Unital Rings

Abstract

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