ac.commutative algebra - Atiyah-MacDonald: exercise 5.29 - "local ring of a valuation ring"

Let B be the subring of K containing A. I am going to prove that B is the localization of A at a prime ideal $Psubset A$, which seems a reasonable interpretation of the statement that B is a local ring of A.

First B is a local ring [by Atiyah-MacDonald Prop 5.18 i)] ; let $M_B$ be its maximal ideal.
Similarly let $M_A$ be the maximal ideal of A.
Define $P=Acap M_B$. Then of course $Psubset M_A$ is prime and if we localize A at P, I claim that we get B:

a) If $pi in A-P$ then $pi notin M_B$ and so $pi$ is invertible in B. Hence for all $ a in A$ we have $a/ pi in B$.
This shows $A_P subset B$.

b) Now we see that B dominates $A_P$: this means we have an inclusion of local rings $A_P subset B$ and of their maximal ideals: $PA_P subset M_B$. But $A_P$ is a valuation ring of K because A is (This is the preceding exercise 28 : these guys know where they are heading!)
and valuations rings are maximal for the relation of domination (Exercise 27: ditto !).
Hence we have the claimed equality $B=A_P$.