Strongly semisimple abelian groups

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For an abelian group G and a ring R, R is a ring on G if the additive group of R is isomorphic to G. G is nil if the only ring R on G is the zero ring, R2 = {0}. G is radical if there is a nonzero ring on G that is radical in the Jacobson sense. Otherwise, G is antiradical. G is semisimple if there is some (Jacobson) semisimple ring on G, and G is strongly semisimple if G is nonnil and every nonzero ring on G is semisimple. It is shown that the only strongly semisimple torsion groups are cyclic of prime order, and that no mixed group is strongly semisimple. The torsion free rank one strongly semisimple groups are characterized in terms of their type, and it is shown that the strongly semisimple and antiradical rank one groups coincide. For torsion free groups it is shown that the property of being strongly semisimple is invariant under quasi-isomorphism and that a strongly semisimple group is strongly indecomposable. Further, for a strongly indecomposable torsion free group G of finite rank, the following are equivalent: (a) G is semisimple, (b) G is strongly semisimple, (c) G≅R+ where R is a full subring of an algebraic number field K such that [K, Q] = rank G where Q is the field of rational numbers and R ≐ Jπ, where π is either empty or an infinite set of primes in K, (d) G is nonnil and antiradical. © 1974 Pacific Journal of Mathematics Manufactured and first issued in Japan.

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... Clearly, faithful groups are nonnil, semi-simple groups are faithful and strongly semi-simple groups are fully faithful. The following theorem (essentially) generalizes Theorem 4.2 of [3]. Proof. ...
... COROLLARY 1.2. (cf. [3], Theorems 2.1, 2.2, Corollary 2.3). An abelian group G is torsion and strongly semi-simple if and only if G = Z(p) for some prime p. ...
... It is clear from this that full faithfulness is a quasi-isomorphism invariant. Using one or the other of the above remarks we get THEOREM 2. ([3]) If G is a torsion free fully faithful abelian group then G is strongly indecomposable. We can get a weak converse to this result as follows. ...
The class of faithful (fully faithful) abelian groups is introduced as a generalization of the semi-simple (strongly semi-simple) groups recently discussed by R. A. Beaumont and D. A. Lawyer. A group is faithful if it admits some associative ring structure with trivial left annihilator. Fully faithful groups are the nonnil groups such that every nontrivial associative ring structure has trivial left annihilator. Several of the results of Beaumont and Lawyer are generalized and it is shown that fully faithful groups arise naturally in classifying strongly indecomposable torsion free groups according to the ring structures they support.
... Зависимость между свойствами кольца и строением его аддитивной группы исследовалась в работах Т. Селе, У. Уиклесса, С. Винсонхалера, Р. Боумонта, Р. Пирса. Л. Фукс сформулировал проблему описания абсолютных радикалов абелевой группы G [4, проблема 94], где под абсолютным радикалом абелевой группы G понимается пересечение радикалов всех ассоциативных колец на G. В [2] поставлена задача описание полупростых абелевых групп, т. е. групп, на которых существуют хотя бы одно ассоциативное кольцо. Изучение абсолютных радикалов абелевой группы и полупростых абелевых групп может быть сведено к случаю редуцированных групп [1]. ...
... В [2] сформулирована проблема описания полупростых групп. Группа называется полупростой, если на ней существует хотя бы одно полупростое ассоциативное кольцо. ...
... Кольцо (A, ×) полупросто, так как является прямой суммой идеалов R i e i (i ∈ I), каждый из которых полупрост [2]. При этом nd × nd = Следовательно, R(G, ×) = 0, т. е. группа G полупроста. ...
E. I. Kompantseva, Rings on almost completely decomposable Abelian groups, Fun-damentalnaya i prikladnaya matematika, vol. 14 (2008), no. 5, pp. 93—101.
... There have been many papers on the general question of which abelian groups support various kinds of ring structures. In addition to the work of Beaumont and coauthors cited previously, see, for example, Wickless ([16]) who motivated Beaumont and Lawver ([3]), who spurred Reid ([10]). Finally, I should like to mention a little bit some work of Nunke since this report does not yet reflect his influence on many of us and the value of his presence among us at the time. ...
Using the theory of spectral subspaces associated with a group of isometries of a Banach space it is proved that each derivation of an AW*-algebra is inner. This constructive method of proof yields a generator b for the case of a skew- adjoint derivation which is seen to be the unique positive generator such that ‖bp‖= δ |Ap| for each central projection p in the AW*-algebra A. © 1974 Pacific Journal of Mathematics Manufactured and first issued in Japan.
Ross Beaumont was born in Dallas, Texas, on July 23, 1914. He grew up in Detroit, Michigan, where his family had moved while he was still young and he received a Bachelor’s degree and a Master of Science degree from the University of Michigan. The year 1940 was a big one in Ross’s life. He received a Ph.D. in Mathematics from the University of Illinois as the first American student of Reinhold Baer, he married and he and his wife Lois moved to Seattle where Ross had been offered a position. According to Lois, neither he (as an adult) nor she “had ever before been west of Chicago”. Evidently the move was a good one personally. Ross served on the faculty at the University of Washington from 1940 until his retirement in 1985. Lois is still a resident of Seattle. Ross and Lois had two children, Linda, now a well known artist in Seattle, and Thomas, now an Assistant Professor in the Department of Mental Health at the University of Minnesota. Ross died on September 28, 1996.
The aim of this paper is to study the correlation between the properties of rings and the structure of their additive groups. In this paper, all multiplications on reduced algebraically compact groups and on divisible torsion-free Abelian groups are described. Absolute Jacobson radicals and absolute nil-radicals in some classes of Abelian groups are found. Using these results, we will give a description of semisimple groups and also of groups on which every ring is a nilpotent ring (a nil-ring, a radical ring) in some concerned classes of Abelian groups.
The present, fourth survey of review articles on Abelian groups includes works reviewed in the years 1979-1984. Also, as in the preceding surveys, no attention has been given to questions involving finite Abelian groups, topological groups, ordered groups, group algebras, modules, or the structure of subgroups, or to questions connected with logic. The word "group" throughout is understood to mean "Abelian group" (except in the case of the group of automorphisms of an Abelian group). Concepts and notation not defined in this survey can be found in the books [117, 118]. The letter Z throughout denotes the group (or ring) of integers, Q the group (or field) of rational numbers, Qp the group (ring) of all rational numbers whose denominator is not divisible by the prime p, Ip the group of all p-adic integers. The torsion part of an Abelian group G is throughout denoted by tG.
The absolute radical of an Abelian group G is the intersection of radicals of all associative rings with additive group G. L. Fuchs formulated the problem on a description of absolute radicals of Abelian groups. For a group from some class of almost completely decomposable Abelian groups the absolute Jacobson radical is described. In the class of almost completely decomposable Abelian groups semisimple groups are described.
An Abelian group is said to be semisimple if it is the additive group of some semisimple associative ring. The problem of describing semisimple groups was formulated by Beaumont and Lawver; later this problem was reduced to the case of reduced groups. In this paper, we describe semisimple groups in the class of countable completely decomposable groups.
This paper is concerned with the ring structures supported by certain mixed abelian groups. A class -M of mixed abelian groups of torsion-free rank one is introduced, and properties of rings on groups in - M are discussed. We provide complete descriptions of the absolute annihilator and the absolute radical of groups in -M These absolute ideals are also investigated for cotorsion groups and reduced algebraically compact groups, thus providing a partial solution to Problem 94 of Fuchs (Infinite abelian groups, Vol. II). The results also allow us to answer a question raised by Rotman (J. Algebra, 9 (1968), 369-387) concerning completions of rings. © 1982, University of California, Berkeley. All Rights Reserved.
Let <G +> be an abelian group. A ring R with additive group <R +> isomorphic to <G +> is a ring on G. G is nil (radical) if and only if R² = (0) (R is nilpotent) for all rings R on G. It is shown that G is a mixed radical group if and only if T is divisible and G/T is radical, where T is the max¬imal torsion subgroup of G. Thus, the study of radical groups is reduced to the torsion free case. A torsion free group G is of field type if and only if there exists a ring R on G such that Q ⊗ R is a field. It is shown that a torsion free group of finite rank is radical if and only if it has no strongly indecomposable component of field type. It follows that finite direct sums of finite rank radical groups are radical. If G is torsion free an element x ∈ G is of nil type if and only if the height vector h(x) = is such that 0 < mi < ∞ for infinitely many i. Multiplications on torsion free groups all of whose nonzero elements are of nil type are discussed under the assumption of three chain conditions on the partially ordered set of types. Two special classes of rank two torsion free radical groups are characterized. An example is given of a torsion free radical group homogeneous of non-nil type, and a simple condition is given for such a homogeneous group to be nonradical.