On the Distributive Law
Wu-Yi Hsiang · Proceedings of the American Mathematical Society · 1960
Introduction.In the theory of rings, the two distributive laws give the only relations between addition and multiplication, and play an important role in the whole theory.The purpose of this paper is to give a full discussion of distributive law in its generalized form.If the following equationholds for all members Xi, y¡ of an algebraic system (R, +, •) with two binary operations, we say that the (m, n)-distributive law holds in (R, +, •), and this identity will be denoted by Dm,n hereafter.Definition.An abelian group R, closed with respect to multiplication, is called an (m, n)-distributive ring if the (m, «)-distributive law holds in (R, +, ■) for fixed m and n.Here, we follow the definition given by Professor R. A. Beaumont in his paper, Generalized rings [l], but we do not require m, n to be greater than or equal to 2. By a ring, we mean a not necessarily associative ring.Thus a ring is an (m, n)-distributive ring for every (m, n); an (m, w)-distributive ring, however, is not necessarily a ring. First FundamentalTheorem.Corresponding to an (m, n)ring (R, +, ■), m, w = 2, there exists a ring (R, +, *) with the same domain and addition, andMoreover, if we take e = 0-0, and X(a) =0-a -0-0, p(a)=a-0 -0-0 for all a in R, then (i) (mn -l)e = 0, (m -l)\(a) =0, (n -l)p(a) = 0. \(a + b) = X(a) + X(b), P(a + b) = p(a) + p(b).Conversely, if (R, +, *) is a ring with an element e and two maps