Abstract
In this paper we study Abelian groups whose homogeneous maps to other Abelian groups are homomorphisms. We consider these groups as modules over the ring of integers and over their endomorphism rings. We also study related issues.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Fuchs, C. J. Maxson, and G. Pilz, “On Rings for Which Homogeneous Maps Are Linear,” Proc. Amer. Math. Soc. 112 (1), 1–7 (1991).
J. Hausen and J. A. Johnson, “Centralizer Near-Rings that are Rings,” J. Austr. Math. Soc. 59, 173–183 (1995).
D. S. Chistyakov and O. V. Lyubimtsev, “AbelianGroups as Endomorphic Modules over their Endomorphism Ring,” Fundam. i Prikl. Mat. 13 (1), 229–233 (2007).
D. S. Chistyakov, “Endomorphic Indecomposable Torsion-Free Abelian Groups of Rank 3,” Vestnik Tomsk. Gos. Univ. Ser. Mat. Mekh. 13 (1), 61–66 (2011).
A. B. van der Merwe, “Unique Addition Modules,” Commun. Algebra 27 (9), 4103–4115 (1999).
O. V. Lyubimtsev and D. S. Chistyakov, “Abelian Groups as UA-Modules over the Ring Z,” Mat. Zametki 87 (3), 412–416 (2010).
D. S. Chistyakov, “Abelian Groups as UA-Modules over their Endomorphism Rings,” Mat. Zametki 91 (6), 934–941 (2012).
A. A. Tuganbaev, Theory of Rings. Arithmetic Modules and Rings (MTsNMO, Moscow, 2009) [in Russian].
P. A. Krylov, A. V. Mikhalev, and A. A. Tuganbaev, Abelian Groups and their Endomorphism Rings (Faktorial Press, Moscow, 2006) [in Russian].
L. Fuchs, Infinite Abelian Groups (Academic Press, New York-London, 1970; Mir, Moscow, 1974), Vol. 1.
L. Fuchs, Infinite Abelian Groups (Academic Press, New York-London, 1973; Mir, Moscow, 1977), Vol. 2.
R. Beaumont and R. Pierce, “Torsion-Free Rings,” Illinois J. Math. 5, 61–98 (1961).
A. A. Fomin and W. Wickless, “Quotient Divisible Abelian Groups,” Proc. Amer. Math. Soc. 126, 45–52 (1998).
O. I. Davydova, “Rank-1 Qutient Divisible Groups,” Fundam. i Prikl.Mat. 13 (3), 25–33 (2007).
A. A. Tuganbaev, “Distributive Rings and Endo-Distributive Modules,” Ukrainsk. Mat. Zhurn. 38 (1), 63–67 (1986).
E. V. Tarakanov, “Endo-Distributive Modules over Dedekind Rings,” in Abelian Groups and Modules (Tomsk. Gos. Univ., Tomsk, 1990), pp. 83–107.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © D.S. Chistyakov, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 2, pp. 61–68.
About this article
Cite this article
Chistyakov, D.S. Homogeneous maps of Abelian groups. Russ Math. 58, 51–57 (2014). https://doi.org/10.3103/S1066369X1402008X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X1402008X