Abstract
The absolute radical of an Abelian group G is the intersection of radicals of all associative rings with additive group G. The problem of describing absolute radicals was formulated by L. Fuchs. He described the absolute Jacobson radical of a torsion Abelian group. In this work, the absolute Jacobson radical and the absolute nil-radical are investigated in some mixed Abelian group classes.
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References
C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York (1962).
L. Fuchs, Infinite Abelian Groups, Vols. 1, 2, Academic Press, New York (1970, 1973).
L. Fuchs and K. M. Rangaswamy, “On generalized regular rings,” Math. Z., 107, 71–81 (1968).
A. V. Ivanov, “Abelian groups with self-injective rings of endomorphisms and with rings of endomorphisms with the annihilator condition,” in Abelian Groups and Modules [in Russian], TSU, Tomsk (1982), pp. 93–109.
N. Jacobson, Structure of Rings, Colloq. Publ., Vol. 37, Amer. Math. Soc. (1956).
E. I. Kompantseva, “Torsion-free rings,” J. Math. Sci., 171, No. 2, 213–247 (2010).
E. I. Kompantseva, “Absolute nil-ideals of Abelian groups,” Fundam. Prikl. Mat., 17, No. 8, 63–76 (2011/2012).
P. A. Krylov, “Mixed Abelian groups as modules over their endomorphism rings,” Fundam. Prikl. Mat., 6, No. 3, 793–812 (2000).
K. M. Rangaswamy, “Abelian groups with endomorphic images of special types,” J. Algebra, 6, 271–280 (1967).
E. H. Toubassi and D. A. Lawver, “Height-slope and splitting length of Abelian groups,” Publ. Mat., 20, 63–71 (1973).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 3, pp. 53–67, 2013.
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Kompantseva, E.I. Abelian dqt-Groups and Rings on Them. J Math Sci 206, 494–504 (2015). https://doi.org/10.1007/s10958-015-2328-2
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DOI: https://doi.org/10.1007/s10958-015-2328-2