
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 formula_1,
where formula_2 for each formula_3 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 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".
Universal property.
Given "R" and "G", there is a ring homomorphism sending each "r" to "r"1 (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 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 , 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 
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"].