Algebra Homework 6

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 that (R[G], +, ∗) is a ring. First, by 2.8.5(a) g∈G Rg is a subgroup of the direct product of (Rg | g ∈ G), and so (R[G], +) is a group. Let (rg)g∈G, (sg)g∈G ∈ R[G]. Then (rg)g∈G + (sg)g∈G = (rg + sg)g∈G addition in R[G] is component-wise = (sg + rg)g∈G addition in R is commutative = (sg)g∈G + (rg)g∈G. So we see that (R[G],+) is an abelian group. Second, we must show that (R[G], ∗) is a semigroup. We know from 3.1.7 that ∗ is well-defined. For (rg)g∈G, (sg)g∈G, (tg)g∈G ∈ R[G], observe (rg)g∈G ∗ ( (sg)g∈G ∗ (tg)g∈G ) = (rg)g∈G ∗  ∑ hl=g shtl   g∈G definition of ∗