Monoid ring

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) = Σ_kl=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

T(r)(1_G) = r, T(r)(g) = 0 for g ≠ 1_G.

where 1_G denotes the identity element in G.

There is also a canonical homomorphism going the other way; the augmentation is the map η_R:R[G] -> R defined by

<math>\sum_{g\in G} r_g g \rightarrow \sum_{g\in G} r_g </math>

The kernel of this homomorphism is called the augmentation ideal and is denoted by J_R(G). It is a free R-module generated by the elements 1 - g, for g in G.

Examples

Given a ring and the monoid of the non-negative integers, N, we obtain the ring of polynomials over that ring.