Monoid Ring - Definition

Definition

Let R be a ring and G be a monoid. Consider all the functions φ : G → R such that the set {g: φ(g) ≠ 0} is finite. Let all such functions be element-wise addable. We can define multiplication by (φ * ψ)(g) = Σ_kl=gφ(k)ψ(l). The set of all such functions φ, together with these two operations, forms a ring, the monoid ring of G over R denoted R. If G is a group, then R denotes the group ring of G over R.

Less rigorously but more simply, an element of R is a polynomial in G over R, hence the notation. We multiply elements as polynomials, taking the product in G of the "indeterminates" and gathering terms:

where r_is_j is the R-product and g_ih_j is the G-product.

The ring R can be embedded in the ring R via the ring homomorphism T : R → R defined by

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

where 1_G is the identity element of G.

There also exists a canonical homomorphism going the other way, called the augmentation. It is the map η_R:R → R, defined by

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