	Let "G" be a group, written multiplicatively, and let "R" be a ring. The group ring of "G" over "R", which we will denote by "R"["G"] (or simply "RG"), is the set of mappings of finite support, where the module scalar product "αf" of a scalar "α" in "R" and a vector (or mapping) "f" is defined as the vector formula_1, and the module group sum of two vectors "f" and "g" is defined as the vector formula_2. To turn the additive group "R"["G"] into a ring, we define the product of "f" and "g" to be the vector