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[G] or RG, is the set of formal sums $\sum_{g \in G} r_g g$ ,

where $r_g \in R$ for each $g \in G$ and r_g = 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 $g \in R[G]$ with the function e_g that maps g to 1 and every other element of G to 0. This way, R[G] is identified with the set of functions {{nowrap|φ: G → R}} such that {{nowrap|{g : φ(g) ≠ 0}}} is finite. equipped with addition of functions, and with multiplication defined by

: $(\phi \psi)(g) = \sum_{k\ell=g} \phi(k) \psi(\ell)$ .

If G is a group, then R[G] is also called the group ring of G over R.

Universal property

Given R and G, there is a ring homomorphism {{nowrap|α: R → R[G]}} sending each r to r1 (where 1 is the identity element of G),

and a monoid homomorphism {{nowrap|β: G → R[G]}} (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 {{nowrap|α': R → S}}, and a monoid homomorphism {{nowrap|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 {{nowrap|γ: R[G] → S}} such that composing α and β with γ produces α' and β

Augmentation

The augmentation is the ring homomorphism {{nowrap|η: R[G] → R}} defined by

: $\eta\left(\sum_{g\in G} r_g g\right) = \sum_{g\in G} r_g.$

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 (additive) monoid of natural numbers N (or {xⁿ} viewed multiplicatively), we obtain the ring R[{xⁿ}] =: R[x] of polynomials over R.

The monoid Nⁿ (with the addition) gives the polynomial ring with n variables: R[Nⁿ] =: R[X₁, ..., X_n].

Generalization

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

References

{{cite book | first = Serge | last = Lang | authorlink=Serge Lang | title = Algebra | publisher = Springer-Verlag | location = New York | year = 2002 | edition = Rev. 3rd | series = Graduate Texts in Mathematics | volume=211 | isbn=0-387-95385-X}}