Groups in rings and in fields

Abstract

The most frequent occurrence of abelian groups, apart from vector spaces, is undoubtedly the groups found in rings and fields. This chapter is devoted to their study. While in general we deal exclusively with associative rings with 1, in the first two sections of this chapter we also include rings without identity as well as not necessarily associative rings (called ‘narings’ for short) in order to make the discussion smoother. As a matter of fact, the collection of narings on a group A displays more pleasant features than the set of associative rings, as demonstrated by the group MultA. This group, suggested by R. Baer, crystallizes the idea of building narings on a group (thus from the additive point of view, associativity in rings seems less natural). The paper devoted to the additive groups of rings, published by Rédei-Szele [1] on the special case of torsion-free rings of rank 1, was the beginning of Szele’s ambitious program on the systematic study of additive groups. Our current knowledge on the additive groups of rings, apart from artinian and regular rings, is still more fragmentary than systematic, though a large amount of material is available in the literature. The inherent problem is that interesting ring properties rarely correspond to familiar group properties. Due to limitation of space, we shall not pursue this matter here; we refer the reader to Feigelstock’s two-volume treatise [Fe]. The problem of rings isomorphic to the endomorphism ring of their additive group (called E-rings) attracted much attention; we present a few miscellaneous results on them. Our final topic concerns multiplicative groups: groups of units in commutative rings and multiplicative groups of fields. While the theory for rings has not reached maturity, there are several essential results inthecase of fields, mostlydue to W. May. Unfortunately, wecannot discuss them here, because they require more advanced results on fields.