# Completion of a polynomial ring w.r.t the homogeneous maximal ideal is the power series ring

I am trying to understand the proof of this fact. On page 183, Eisenbud defines a map from the formal power series ring $S[[x_1,x_2,...,x_n]]$ to $R/m^i$ where $R=S[x_1,x_2,...,x_n]$ sending $f$ to $f+m^i$. I have trouble understanding this map since, $f$ need not be an element of $R$ and so what does the coset $f+m^i$ mean? Thanks for any help.

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Pick the series $f$ and just drop all terms of degree larger that $i$. That is a polynomial, the image under Eisenbud's map. – Mariano Suárez-Alvarez Feb 1 '11 at 2:26
Thanks. Seems like a strange notation. – Dev Bappa Feb 1 '11 at 23:20
it is simple! all monomials of degree >=i lies in m^i – user20970 Dec 10 '11 at 10:40