Semisimple rings on completely decomposable Abelian groups. (English. Russian original) Zbl 1166.20048

J. Math. Sci., New York 154, No. 3, 324-332 (2008); translation from Fundam. Prikl. Mat. 13, No. 3, 69-80 (2007).
The author characterizes the countable completely decomposable Abelian groups which admit a semisimple ring multiplication. The characterization is in terms of the order relation of the types of the rank 1 direct summands and the number of summands of each type.


20K20 Torsion-free groups, infinite rank
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
20K25 Direct sums, direct products, etc. for abelian groups
20M25 Semigroup rings, multiplicative semigroups of rings
Full Text: DOI


[1] R. A. Beaumont and D. A. Lawver, ”Strongly semisimple Abelian groups,” Publ. J. Math., 53, No. 2, 327–336 (1974). · Zbl 0315.20047
[2] R. A. Beaumont and R. S. Pierce, ”Torsion-free rings,” Illinois J. Math., 5, 61–98 (1961). · Zbl 0108.03802
[3] L. Fuchs, Infinite Abelian Groups, Vol. I, Academic Press, New York (1970). · Zbl 0209.05503
[4] L. Fuchs, Infinite Abelian Groups, Vol. II, Academic Press, New York (1973). · Zbl 0257.20035
[5] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ., Vol. 37, Amer. Math. Soc., Providence (1956).
[6] E. I. Kompantseva, ”Semisimple nonreduced torsion-free Abelian groups,” in: Abelian Groups and Modules, No. 13–14, Tomsk (1996), pp. 67–76. · Zbl 1061.20510
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.