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