Power series
Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid "N" used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of "N", the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from "N" to a ring "R" with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as the ring completion of the polynomial ring with respect to the ideal generated by .