Group Ring - Definition

Definition

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, is the set of mappings f : G → R of finite support, where the product of a scalar in R and a vector (or mapping) f is defined as the vector, and the sum of two vectors f and g is defined as the vector . To turn the commutative 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":

or simply

where this doesn't cause confusion.