in algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. as a free module, its ring of scalars is the given ring, and its basis is one-to-one with the given group. as a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.