# 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 ,
where for each 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 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 *R*[*G*] is also called the group ring of *G* over *R*.

## Universal property

Given *R* and *G*, there is a ring homomorphism α: *R* → *R*[*G*] sending each *r* to *r*1 (where 1 is the identity element of *G*),
and a monoid homomorphism β: *G* → *R*[*G*] (where the latter is viewed as a monoid under multiplication) sending each *g* to 1*g* (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: *R*[*G*] → *S* such that composing α and β with γ produces α' and β
'.

## Augmentation

The **augmentation** is the ring homomorphism *η*: *R*[*G*] → *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.

## Examples

Given a ring *R* and the (additive) monoid of natural numbers **N** (or {*x*^{n}} viewed multiplicatively), we obtain the ring *R*[{*x*^{n}}] =: *R*[*x*] of polynomials over *R*.
The monoid **N**^{n} (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*].

## See also

## References

- Lang, Serge (2002).
*Algebra*. Graduate Texts in Mathematics.**211**(Rev. 3rd ed.). New York: Springer-Verlag. ISBN 0-387-95385-X.

## Further reading

- R.Gilmer.
*Commutative semigroup rings*. University of Chicago Press, Chicago–London, 1984