Group ring

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. 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. A group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. 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 (or simply RG), is the set of mappings f : G → R 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 x ↦ α ⋅ f ( x ) {displaystyle xmapsto alpha cdot f(x)} , and the module group sum of two vectors f and g is defined as the vector x ↦ f ( x ) + g ( x ) {displaystyle xmapsto f(x)+g(x)} . To turn the additive group R into a ring, we define the product of f and g to be the vector The summation is legitimate because f and g are of finite support, and the ring axioms are readily verified. Some variations in the notation and terminology are in use. In particular, the mappings such as f : G → R are sometimes written as what are called 'formal linear combinations of elements of G, with coefficients in R':