The following is taken from the introduction paragraph of the volume Commutative Algebra. The equation mentioned below with $(*)$ is the following:
Diophantine System $(*)$: $P(x_1,...,x_n)=0$.
The part I do not quite understand is the following:
The original problem (to solve $(*)$ in $\mathbb Z$), thus weakened, is finally transformed into the analogous problem for local rings of the type $A/\mathfrak m^n$.
How is the problem of solving $(*)$ in $\mathbb Z$ weakened into solving in congruences modulo prime powers in $\mathbb Z/p^k\mathbb Z$? And how does solving $(**)$ above give a solution in the completion $\widehat{\mathbb Z}_{(p)}$?