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 R[G] or RG, is the set of formal sums , where for each and r<sub>g</sub> = 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, R[G] 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 R[G] and the right-hand side is understood in G.
Alternatively, one can identify the element with the function e<sub>g</sub> that maps g to 1 and every other element of G to 0. This way, R[G] is identified with the set of functions such that } is finite. equipped with addition of functions, and with multiplication defined by
If G is a group, then R[G] is also called the group ring of G over R.
Given R and G, there is a ring homomorphism sending each r to r1 (where 1 is the identity element of G), and a monoid homomorphism (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 ñ(r) commutes with ò(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 , and a monoid homomorphism to the multiplicative monoid of S, such that ñ'(r) commutes with ò'(g) for all r in R and g in G, there is a unique ring homomorphism such that composing ñ and ò with ó produces ñ' and ò '.
The augmentation is the ring homomorphism defined by
The kernel of η is called the augmentation ideal. It is a free R-module with basis consisting of 1 â g for all g in G not equal to 1.
Given a ring R and the (additive) monoid of natural numbers N (or {x<sup>n</sup>} viewed multiplicatively), we obtain the ring R[{x<sup>n</sup>}] =: R[x] of polynomials over R. The monoid N<sup>n</sup> (with the addition) gives the polynomial ring with n variables: R[N<sup>n</sup>] =: R[X<sub>1</sub>, ..., X<sub>n</sub>].
If G is a semigroup, the same construction yields a semigroup ring R[G].