Let *R* be a ring and *G* be a monoid.
We can look at all the functions φ : *G* -> *R* such that the
set {*g*: φ(*g*) ≠ 0} is finite. We can define addition
of such functions to be element-wise additions. We can define multiplication
by
(φ * ψ)(*g*) = Σ_{kl=g}φ(*k*)ψ(*l*).
The set of all these functions, together with these two operations, forms a ring, the **monoid ring** of *R* over *G*; it is denoted by *R*[*G*].
If *G* is a group, then it is called the **group ring** of *R* over *G*.

The ring *R* can be embedded into the ring *R*[*G*] via the ring homomorphism *T*: *R*->*R*[*G*] defined by

*T*(*r*)(1_{G}) =*r*,*T*(*r*)(*g*) = 0 for*g*≠ 1_{G}.

There is also a canonical homomorphism going the other way; the **augmentation** is the map *η _{R}*: