I have to solve such a system of equations:

$\displaystyle a_{11}x_1 + a_{21}x_2 + \ldots + a_{n1}x_n =y_1 (mod \, p^k)$

$\displaystyle a_{12}x_1 + a_{22}x_2 + \ldots + a_{n2}x_n =y_2 (mod \, p^k)$

$\displaystyle \ldots$

$\displaystyle a_{1n}x_1 + a_{2n}x_2 + \ldots + a_{nn}x_n =y_n (mod \, p^k)$

p is a prime, k = 1,2,3...

When k=1 I can use Gauss elimination method, but it doesn't work when k>1 Any ideas, links, pdf's about such equations?