The complete description of the square subgroup of a torsion abelian group and an elementary construction of a mixed abelian group (A, +, 0), such that the quotient group of A modulo the square subgroup □A is not a nil-group, are given (also for the associative case). Some A. M. Aghdam's results concerning square subgroups for the associative case are proven. The relationship between square subgroups for both the cases of associative and general rings is partially investigated.

Some New Results for the Square Subgroup of an Abelian Group

Abstract Almost complete description of abelian groups (A, +, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

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A Note on Additive Groups of Some Specific Associative Rings

This paper presents new results concerning the structure of \(SI\)-groups, and refines and purifies the results obtained in this field by Shalom Feigelstock ['Additive groups of rings whose subrings are ideals' Bull. Austral. Math. Soc. \(55\) (1997), 477-481]. The structure theorem describing torsion-free \(SI\)-groups is proved in the associative case. Numerous examples of \(SI\)-groups are given. Some inconsistencies in Feigelstock's article are noted and removed.

DOI:

 10.1017/S0004972714000641

On SI -groups

The first example of a torsion-free abelian group such that the quotient group of modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen 27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from to is presented. Ring multiplications on -pure subgroups of the additive group of the ring of -adic integers are investigated using only elementary methods.

A torsion-free abelian group exists whose quotient group modulo the square subgroup is not a \(nil\)-group

The notion of the square submodule of a module M over an arbitrary commutative ring R; which is denoted by RM; was introduced by Aghdam and Najazadeh in [3]. In fact, RM is the R submodule of M generated by the images of all bilinear maps on M: Furthermore, given a submodule N of an R module M; we say that M is nil modulo N if (M M) N for all bilinear maps on M: The main question about the square submodule is that whether the quotient module M= M is a nil module? In this paper, we investigate the square submodules of some classes of modules over commutative domains. Then, we have some results related to splitting modules which we need in our discussions. Finally, we give some examples of mixed Abelian groups A such that the quotient groups A= A are not nil.

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On the Square Submodule of a Mixed Module

Let R be a commutative ring with identity, and M an R-module. We introduce square submodule, □M, of M as a generalization of the square subgroup in the sense of [9] and investigate its properties over a commutative domain.

Alireza Najafizadeh

Papers

Square Submodule of a Module