  Let $n; r; e; s$ be are positive integers and the prime p; the finite local
principal ideals ring of parameters $p; n; r; e; s)$ $GR(p^n;r)[x]/(x^e - pu ;
x^s),$ is defined by an invertible element u of the Galois ring $GR(p^n; r)$ of
characteristic $p^n$ of order $p^{nr}.$ It is called Galois-Eisenstein ring of
parameters $(p; n; r; e; s)$. A basic problem, which seems to be very difficult
is to determine all non-isomorphism pure Galois-Eisenstein rings of parameters
$(p; n; r; e; s).$ In this paper, this isomorphism problem for pure
Galois-Eisenstein rings of parameters $(p; n; r; e; s)$ is investigated.
