Group Ring

Notation

$ G(\cdot) $: a group.

$R(\hat{+}, \hat{\times})$: a ring.

$ 0_R $: additive identity (a.k.a. zero) of $ R $.

$ RG(+, \times) = $$\{ \sum_{g \in G} r_g g \} $ where $ \{ r_g \neq 0_R \} $ is finite: group ring from $ G $ and $ R $.

Operations

$ \sum_i r_i g_i + \sum_j r_j g_j = $$\sum_k (r_i \hat{+} r_j) g_k $ where $ g_i = g_j = g_k $.

$ \sum_i r_i g_i \times \sum_j r_j g_j = $$\sum_{i,j} (r_i \hat{\times} r_j) (g_i \cdot g_j )$.