CHAPTER XII - THE ADDITIVE GROUP OF RINGS

Abstract

This chapter discusses the additive group of rings. The general problem of constructing rings on a given group is much more complicated for torsion free and mixed groups than in the torsion case. There is a certain class of torsion free groups that are no direct sums of cyclic groups but for which the results may be extended without any change; these are the torsion free divisible groups. Every group appears as the additive group of some ring, that is, every group may be made into a ring with trivial multiplication by defining the product of every pair of its elements to be equal to zero. It is not difficult to describe all rings whose additive group is a direct sum of cyclic groups, in terms of integers satisfying certain postulates, but the conditions concerning the isomorphism of two rings with the same additive group are too complicated to discuss. It turns out that the problem on arbitrary torsion groups is not much more difficult than that for direct sums of cyclic groups, as the knowledge of the products of the basis elements of some basic subgroup already uniquely determines the product of an arbitrary pair of elements.