On Proving the Absence of Zero-Divisors for Semi-Group Rings

Abstract

For any semi-group S and any ring Λ with unit 1 (always taken to be distinct from 0, the neutral element of Λ under addition) there is known to exist a ring Λ[S] ⊇S which is a A-bimodule such that (i) S is a sub semi-group of the multiplicative semi-group of Λ[S], (ii) λs = sλ, (iii) λ(st) = (λs)t = s(λt) ( s, t ∊ S and λ∊Λ) and (iv) Sis a Λ -basis of Λ[S]. This ring is uniquely determined by these conditions and is usually called the semi-group ring of S over Λ. It may be described explicitly as consisting of the functions f: S → Λ which vanish at all but finitely many places, with functional addition (f+g) (s) = f(s) + g(s) and convolution (fg) (s) = Σf(u) g(v) (uv = s) as the ring operations, the functional A-bimodule operations (λf) (s) = λf(s) and (fλ) (s) - f(s)λ, and each s ∊ S identified with the characteristic function of { s} with values in Λ.