Abstract A hypothesis stated in [16] is confirmed for the case of associative rings. The answers to some questions posed in the mentioned paper are also given. The square subgroup of a completely decomposable torsion-free abelian group is described (in both cases of associative and general rings). It is shown that for any such a group A, the quotient group modulo the square subgroup of A is a nil-group. Some results listed in [16] are generalized and corrected. Moreover, it is proved that for a given abelian group A, the square subgroup of A considered in the class of associative rings, is a characteristic subgroup of A.

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A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

Abstract It is studied how rank two pure subgroups of a torsion-free Abelian group of rank three influences its structure and type set. In particular, the criterion for such a subgroup B to be a direct summand of a torsion-free Abelian group of rank three with the finite type set containing the greatest element which does not belong to the type set of B, is presented. Some results for nil groups and the square subgroup of a decomposable torsion-free Abelian group are also achieved. Moreover, new results for mixed Abelian groups supporting only associative rings are obtained. In particular, the first example of an Abelian group supporting only associative rings but not only commutative rings is given.

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A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings

Abstract The paper contains a simple solution of Stratton and Webb’s problem related to the so-called absolute annihilators of abelian groups researched also by Feigelstock. Namely, there are provided constructions of indecomposable torsion-free abelian groups G of any finite rank greater than two for which the quotient group modulo the subgroup generated by squares of all possible rings on G supports a non-zero associative ring structure. They are much less complicated than those obtained before and show that groups satisfying the mentioned condition are not as rare phenomena as might be supposed. Moreover, most of these constructions led to new examples of CR- and AR-groups, i.e. abelian groups supporting only commutative and only associative rings, respectively. In particular, the first examples of indecomposable as well as decomposable AR-groups which do not satisfy the condition CR are presented. The first example of an indecomposable torsion-free group satisfying the condition CR only in the class of associative rings is also obtained.

A simple solution of Stratton and Webb’s problem

Let G be a torsion free abelian group of rank two. The principal purpose of the present paper is to show the structure of a ring on G. We prove that if G is indecomposable with cardinality of the typeset two or three, then all of the rings on G are associative and commutative. It will be presented two examples, one about an associative ring, and other non-associative ring on the torsion-free abelian groups of rank two. Mathematics Subject Classification: 20K15

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A. M. Aghdam

Papers

Rings on indecomposable torsion free groups of rank two