group ring

{{algebra}} Given ring ''R'' with identity not equal to zero, and group <math>G = \{g_1, g_2, ..., g_n\}</math>, the group ring ''RG'' has elements of the form <math> a_1 g_1 + a_2 g_2 + ... + a_n g_n </math> (where <math>a_i \isin R </math>) such that the sum of <math> a_1 g_1 + a_2 g_2 + ... + a_n g_n </math> and <math> b_1 g_1 + b_2 g_2 + ... + b_n g_n </math> is <math> (a_1 + b_1) g_1 + (a_2 + b_2) g_2 + ... + (a_n + b_n) g_n</math> and the product is <math> \sum_{k=1}^n \left ( \sum_{g_i g_j = g_k} a_i b_j \right ) g_k </math>.