The group algebra is the set of all formal expressions

(1)

where
for all
and
for all but finitely many indices so that for sufficiently large (say, ). Hence, we can write the general element as

(2)

Assigning

(3)

defines an isomorphism of -algebras between and the polynomial ring .

More generally, if
is the subsemigroup of generated by the elements , for , the semigroup algebra is isomorphic to the subalgebra
of the polynomial ring generated by the monomials