Suppose R is a ring and X1, . . . , Xn are variables. Show that for any subset, even the empty set, S ? {X1a1 ? ? ? Xnan | a1, . . . , an ? Z} the R-subalgebra of the Laurent polynomial ring

R[X1 1 , . . . ,Xn 1 ], generated by S, e.g. R[S], is isomorphic as an R-algebra to the monoid algebra R[M] for an appropriate monoid M.

We want to show that the R-subalgebra of the Laurent polynomial ring, generated by S, is isomorphic as an R-algebra, to the monoid algebra R[M] for some monoid.