Abelian Groups - László Fuchs [Springer]

Chapter 18
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 Mult A. 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 in the case of fields, mostly due to W. May. Unfortunately, we cannot discuss them
here, because they require more advanced results on fields.
1 Additive Groups of Rings and Modules
In this section we collect a few fundamental facts on the additive groups of rings.
But first of all, conventions on the terminology are in order.
Rings and Their Additive Groups In this chapter, by a ring we shall mean
an associative ring (with or without identity), and by a naring a not necessarily
associative ring (there are many of those of importance, like Lie rings, alternative
rings, etc.). As customary, we will attribute to the ring properties of its underlying
additive group in cases when no confusion may arise. Consequently, terms like p-
ring, torsion or torsion-free ring, divisible and reduced ring, pure ideal, etc. will
make perfectly good sense without any further comment. If desirable, distinction
will be made between the ring R and its additive group R C ;however,extreme
caution is necessary in the context of direct decompositions. A ring R whose
© Springer International Publishing Switzerland 2015
L. Fuchs, Abelian Groups, Springer Monographs in Mathematics,
DOI 10.1007/978-3-319-19422-6_18
673