Monoid ring


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.

Definition

Let R be a ring and let G be a monoid. The monoid ring or monoid algebra of G over R, denoted R 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, R is 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 is also called the group ring of G over R.

Universal property

Given R and G, there is a ring homomorphism sending each r to r1,
and a monoid homomorphism sending each g to 1g.
We have that α commutes with β 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 α' commutes with β' for all r in R and g in G, there is a unique ring homomorphism such that composing α and β with γ produces α' and β
'.

Augmentation

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 – g for all g in G not equal to 1.

Examples

Given a ring R and the monoid of natural numbers N, we obtain the ring R =: R of polynomials over R.
The monoid Nn gives the polynomial ring with n variables: R =: R.

Generalization

If G is a semigroup, the same construction yields a semigroup ring R.