Let R be a ring and G be a monoid. We can look at all the functions φ : G -> R such that the set {g: φ(g) ≠ 0} is finite. We can define addition of such functions to be element-wise additions. We can define multiplication by (φ * ψ)(g) = Σkl=gφ(k)ψ(l). The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G; it is denoted by R[G]. If G is a group, then it is called the group ring of R over G.
The ring R can be embedded into the ring R[G] via the ring homomorphism T: R->R[G] defined by
There is also a canonical homomorphism going the other way; the augmentation is the map ηR:R[G] -> R defined by