1 Let (R, +, ·) be a ring and G a semigroup. Throughout this problem let h, l, m, n vary in G unless clearly fixed. From 3.1.7, we have the semigroup ring (R[G], +, ∗) of G over R with the well-defined multiplicative operation ∗ : R[G]×R[G] → R[G], (rg)g∈G ∗ (sg)g∈G → (tg)g∈G, where tg = ∑ hl=g rh · sl. Note that we are using ∗ to emphasize when the multiplication is in R[G]. (a) We will show t...