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 elementwise additions. We can define multiplication by (φ * ψ)(g) = Σ_{kl=g}φ(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
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
The kernel of this homomorphism is called the augmentation ideal and is denoted by J_{R}(G). It is a free Rmodule generated by the elements 1  g, for g in G.
Given a ring and the monoid of the nonnegative integers, N, we obtain the ring of polynomials over that ring.
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