A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: . A set for which addition makes sense (is closed and associative) is called a monoid. The set of functions from a monoid "N" to a ring "R" which are nonzero at only finitely many places can be given the structure of a ring known as "R"["N"], the monoid ring of "N" with coefficients in "R". The addition is defined component-wise, so that if , then for every "n" in "N". The multiplication is defined as the Cauchy product, so that if , then for each "n" in "N", "c""n" is the sum of all "a""i""b""j" where "i", "j" range over all pairs of elements of "N" which sum to "n".