I have some question about the propositon(c) of thetheorem 3.16 in Qing Liu's book.
the statement is:
Let (A,m) be a Noetherian local ring, and $\hat{A}$ its m-adic completion.
Let (B,n) be a local ring such that $A\subseteq B \subseteq \hat{A}$ and mB =n. Then the n-adic completion $\hat{B}$ is isomorphic to $\hat{A}$.
the proof of c is:
Let n $\geq$ 1. We have $n^n=m^nB$.Since the composition $A/m^n \to B/m^nB \to \hat{A}/m^n\hat{A}$ is an isomorphism, $B/m^nB \to \hat{A}/m^n\hat{A}$ is surjective. It remains to show that it is injective; that is, that $m^n\hat{A}\cap B=m^nB$. We have $B=A+mB=A+m^2B=\dots=A+m^nB$.
I only want to know why B=A+mB. Can someone help me? [1]: https://i.stack.imgur.com/Cr1Hs.png [2]: https://i.stack.imgur.com/bsAtJ.png
