In abstract algebra, a monoid ring is a new ring constructed from some other ring and a monoid.
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 elementwise addable. We can define multiplication by (φ * ψ)(g) = Σ_{kl=g}φ(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[G]. If G is a group, then R[G] denotes the group ring of G over R.
Less rigorously but more simply, an element of R[G] 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_{i}s_{j} is the Rproduct and g_{i}h_{j} is the Gproduct.
The ring R can be embedded in the ring R[G] via the ring homomorphism T : R → R[G] defined by
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[G] → R ,defined by
The kernel of this homomorphism, the augmentation ideal, is denoted by J_{R}(G). It is a free Rmodule generated by the elements 1  g, for g in G.
Examples
Given a ring R and the (additive) monoid of the natural numbers N (or {x^{n}} viewed multiplicatively), we obtain the ring R[{x^{n}}] =: R[x] of polynomials over R. The Monoid N^{n} (with the addition) gives the polynomial ring with n variables: R[N^{n}] =: R[X_{1}, ..., X_{n}].
References
 Lang, Serge (2002). Algebra. Graduate Texts in Mathematics 211 ((Rev. 3rd ed.) ed.). New York: Springer. ISBN 038795385X.
Full article ▸
