Monoid ring explained

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_gr_gg

,where

r_g\inR

for each

g\inG

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\inR[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 such that 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 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 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, and a monoid homomorphism 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 such that composing α and β with γ produces α' and β'.

Augmentation

The augmentation is the ring homomorphism defined by

η\left(\sum_g\inr_gg\right)=\sum_g\inr_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 viewed multiplicatively), we obtain the ring R[{''x''''n''}] =: R[''x''] of polynomials over R.The monoid Nⁿ (with the addition) gives the polynomial ring with n variables: R['''N'''''n''] =: R[''X''1, ..., ''X''''n''].

Generalization

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

References

Book: Lang, Serge . Serge Lang

. Serge Lang . Algebra . . New York . 2002 . Rev. 3rd . . 211 . 0-387-95385-X.