On the Theory of Characters of Groups and Semi-Groups in Normed Vector Rings.

1. Recently I. Gelfand1 has defined a class of characters of Abelian groups embedded in a normed commutative vector ring. The present note contains an elaboration of this theory with a strengthening of the main theorem and extensions to semi-groups. Let 9R be a normed commutative vector ring with unit element e, £ the set of maximal ideals 9 in 9?, 9/9) the corresponding residue class rings each of which is a field. Defining the norm lXII of a residue class X, modulo 91, as min I |x| for x e X, Gelfand shows that 9/9 is a normed field and hence isomorphic with the complex field. Let Xa be the class of elements congruent to ae (mod. 9), a arbitrary complex number. Then UXa = Wand Ka. +X, = X,.+ p,XaXX = Xp. Define2