Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings

We next want to construct a much larger ring in which infinite sums of multiples of elements of S are allowed. In order to insure that multiplication is well-defined, from now on we assume that S has the following additional property: (#) For all s ∈ S, {(s1, s2) ∈ S × S : s1s2 = s} is finite. Thus, each element of S has only finitely many factorizations as a product of two elements. For example, we may take S to be the set of all monomials {x1 1 · · · xn n : (k1, . . . , kn) ∈ N} in n variables. For this chocie of S, the usual semigroup ring R[S] may be identified with the polynomial ring R[x1, . . . , xn] in n indeterminates over R.