MATH 415 Lecture 23

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Exam Discussion

$ϕ ((m, n)) = (m + n, 0)$ and $ψ ((m, n)) = (0, n)$ are elements of $E n d (⟨ ℤ \times ℤ, + ⟩)$

Let $⟨ F [x], + ⟩$ be the additive group of $F [x]$ . Let $ϕ : p (x) \mapsto x p (x)$ and $ψ : p (x) \mapsto \frac{d p}{d x}$

Thus we have $ψ (ϕ (p (x))) - ϕ (ψ (p (x))) = 1$

Hence $⟨ ϕ, ψ ⟩ \subset E n d (F [x])$ .

We start with a group $G = {g_{i} ∣ i \in I}$ . Let $R$ be a commutative ring with nonzero unity.

R G = {\sum_{i} a_{i} g_{i} ∣ a_{i} \in R, g_{i} \in G, all but finite number of a_{i} are 0}

We define addition of elements of $R G$ as

(\sum_{i} a_{i} g_{i}) + (\sum_{i} b_{i} g_{i}) = \sum_{i} (a_{i} + b_{i}) g_{i}

We define multiplication of elements of $R G$ as

(\sum_{i} a_{i} g_{i}) (\sum_{i} b_{i} g_{i}) = \sum_{i} (\sum_{g_{j} g_{k} = g_{i}} a_{j} b_{k}) g_{i}

(similar to multiplication of polynomials)

$⟨ R G, +, \cdot ⟩$ is a ring. In particular, it is called a group ring of $G$ over $R$ .

If $R = F$ is a field, then $⟨ F G, +, \cdot ⟩$ is a group algebra of $G$ over $F$ .