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