In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.
Let R be a ring and let G be a monoid. The monoid ring or monoid algebra of G over R, denoted RG] or RG, is the set of formal sums , where for each and rg = 0 for all but finitely many g, equipped with coefficient-wise addition, and the multiplication in which the elements of R commute with the elements of G. More formally, RG] is the free R-module on the set G, endowed with R-linear multiplication defined on the base elements by g·h := gh, where the left-hand side is understood as the multiplication in RG] and the right-hand side is understood in G.
Alternatively, one can identify the element with the function eg that maps g to 1 and every other element of G to 0. This way, RG] is identified with the set of functions φ: G → R such that {g : φ(g) ≠ 0} is finite. equipped with addition of functions, and with multiplication defined by
If G is a group, then RG] is also called the group ring of G over R.
Given R and G, there is a ring homomorphism α: R → RG sending each r to r1 (where 1 is the identity element of G), and a monoid homomorphism β: G → RG (where the latter is viewed as a monoid under multiplication) sending each g to 1g (where 1 is the multiplicative identity of R). We have that α(r) commutes with β(g) for all r in R and g in G.
The universal property of the monoid ring states that given a ring S, a ring homomorphism α': R → S, and a monoid homomorphism β': G → S to the multiplicative monoid of S, such that α'(r) commutes with β'(g) for all r in R and g in G, there is a unique ring homomorphism γ: RG] → S such that composing α and β with γ produces α' and β '.
The augmentation is the ring homomorphism η: RG] → R defined by
The kernel of η is called the augmentation ideal. It is a free R- module with basis consisting of 1 – g for all g in G not equal to 1.
Given a ring R and the (additive) monoid of natural numbers N (or {xn} viewed multiplicatively), we obtain the ring R[{xn}] =: Rx] of polynomials over R. The monoid Nn (with the addition) gives the polynomial ring with n variables: RNn] =: RX1, ..., Xn].
If G is a semigroup, the same construction yields a semigroup ring RG].