Polynomials

Let R be a ring, and let N+ be the monoid of the non-negative integers, with addition as the monoid composition. We consider the ring R[N+] whose elements are the maps from N+ to R, and whose multiplication is the convolution, as defined in Section II.7. Let x denote the element of R[N+] that takes the value 1 R (the identity element of R) at the element 1 of N+ and the value 0 R at every other element of N+. Then every element f of R[N+] may be viewed as a formal power series in x, the coefficient of x n being f(n). More precisely, the purely formal infinite “sum” $$ \sum\nolimits_{{n \geqq 0}} {f(n){x^n}} $$ may be regarded as an element of R [N+] in the evident way, merely by observing that, at each element m of N+, all the summands f(n)x n in which n is different from m take the value 0 R , while f(m)x m takes the value f(m) at m. In this sense, we have $$ f = \sum\limits_{{n \geqq 0}} {f(n){x^n}} $$ and the multiplication in R[N+] is determined by R-linearity and the fact that x p x q = x p+q for all non-negative exponents p and q. When we have this description in mind, we write R[[x]] for R[N+], and we refer to it as the ring of formal power series over R. Usually, we shall be in a situation where R is commutative, and we note that then R[[x]] is actually an R-algebra, in the sense used before with a field in the role of R.