Addytywne grupy pierścieni łącznych

This doctoral dissertation deals with issues related to additive groups of rings, mainly associative ones. Neither unitarity nor commutativity of rings is assumed. The main goal of creating this work was the continuation of research on the impact of the additive structure of the ring on its multiplication, which starting point can be dated back to the end of the rst half of the 20th century. It was motivated by a multitude of natural questions related to this subject, which are often closely connected with the associativity of rings constructed on abelian groups. An additional idea playing an important role in creating this dissertation was the concept of comparing results obtained in both cases of associative and not necessarily associative rings. In this thesis, the classical issues concerning nil-groups (i.e., abelian groups admitting only trivial ring structure) and their contemporary generalizations such as the concept of the square subgroup of an abelian group (i.e., the group generated by squares of all possible rings with the xed additive group) were considered. Abelian groups A whose structure determines typical ring properties in each ring with the additive group A were also studied. They included: commutativity, associativity and commutativity implied by associativity. Moreover, the structure of abelian groups A was investigated in cases where every ring R on A satis es the following conditions: every subring of R is an ideal in R, every subgroup of A is a subring of R, every subgroup of A is an ideal in R, the relation of being an ideal in R is transitive, i.e., all ideals of any ideal of R are ideals in R. The greatest impact on the scope of the issues examined in this work has been made by research articles written by R. A. Beaumont, R. J. Wisner, S. Feigelstock, A. M. Aghdam and A. Naja zadeh. The methodology of research related to the results presented in this work is partially based on classical tools of constructing ring multiplications on abelian groups such as the tensor product of abelian groups and types theory. Due to their limited usefulness in the description of additive groups of associative rings, non-standard and new tools were used frequently. Sometimes also elementary methods of proving were used, especially in situations where proofs of classical theorems were simpli ed or well-known results were clari ed and complemented. The dissertation is divided into six chapters. The rst one is an introduction to the topic. Therefore, it contains numerous auxiliary results that are used in the main part of this work. One of the most important preliminary results is a characterization of p-pure subgroups of the additive group of the ring of p-adic integers. It has been obtained by using only elementary methods and it turned out to be crucial in the rst constructions of torsion-free abelian groups for which the quotient group modulo the square subgroup is not a nil-group (even in the case of considering only associative rings). They were placed in the second chapter whose main results are also as follows: considerable simpli cation and signi cant generalization of the analogous result for mixed abelian groups obtained recently by A. Naja zadeh, descriptions of square subgroups of any torsion abelian group and any completely decomposable torsion-free abelian group, generalizations of numerous important results obtained in this eld by A. M. Aghdam. The third chapter deals with (A)CR-groups, i.e., abelian groups A satisfying the condition: if R is any (associative) ring with the additive group A, then R is a commutative ring. It contains detailed characterizations of torsion-free (A)CR-groups of rank two which completes and orders current knowledge of these groups, the description of all torsion-free ACR-groups of rank two which are not CR-groups, a partial characterization of mixed ACR-groups and the description of almost all relationships between conditions CR, ACR and AR for torsion, torsion-free and mixed groups, where the last condition means that every ring on the xed abelian group is associative. 1 In the fourth chapter the results concerning SI-groups are presented. SI-groups are abelian groups A such that every (not necessarily associative) ring R with the additive group A has the property that each subring of R is an ideal. In the chapter, errors related to subtle considerations connected with associativity of rings constructed by S. Feigelstock in the paper Additive groups of rings whose subrings are ideals [Bull. Austral. Math. Soc. 55: 477 481] were corrected. As a consequence, a more general concept of an SIH-group was introduced. Such a group satis es the condition SI restricted to the class of associative rings. Moreover, new results for non-torsion SI(H) -groups were presented. They included: classi cation theorems and the construction of a non-splitting SI(H) -group for which the quotient group modulo the square subgroup is not a nil-group in the class of associative rings (a mixed abelian group A is called non-splitting if its torsion part is not a direct summand of A). Furthermore, it was proved that the class of SIHgroups is a proper subclass of the class od ACR-groups and the class of SI-groups is a proper subclass of the class of CR-groups. The fth chapter is devoted to the description of T I-groups which are a natural generalization of SIH-groups. Namely, abelian groups A satisfying the condition: if R is any associative ring with the additive group A, then R is a lial ring, are considered. The liality of a ring R means that the ring R is associative and all ideals of any ideal of R are ideals in R. Unexpected applications of lial rings were revealed during the study of the square subgroup of a mixed SIgroup and torsion-free CR-groups. This chapter contains classi cations of torsion T I-groups and torsion-free T I-groups which are not nil-groups in the class of associative rings. It also includes the description of the structure of the torsion part of a mixed T I-group, which turned out to be identical to that of an SI(H) -group, and examples of mixed T I-groups which are not SIH-groups. Moreover, relationships between the conditions T I, ACR and CR were investigated. In the sixth chapter, the classi cation of abelian groups A such that every subgroup of A is a subring of each associative ring with the additive group A, is given up to the structure of torsion-free nil-groups in the class of associative rings. These groups are called S-groups. Surprisingly, the obtained description is also the classi cation of abelian groups A satisfying the condition: if R is an associative rings on A, then all subgroups of A are ideals in R. In the opinion of the author of this doctoral dissertation, the most important results from those above-mentioned are constructions of abelian groups for which the quotient group modulo the square subgroup is not a nil-group in the class of associative rings. They provide an answer to the question posed by A. E. Stratton and M. C. Webb in 1980 concerning the existence of such groups. It was motivated by previous research related to so-called absolute annihilators of abelian groups modulo some xed subgroups conducted by S. Feigelstock. In particular, one of the aforementioned constructions had led to new examples of non-splitting groups which have been investigated relatively poorly until now.