Non isomorphic pure Galois-Eisenstein rings

Abstract: Let $n; r; e; s$ be are positive integers and the prime p; the finite local principal ideals ring of parameters $p; n; r; e; s)$ $GR(p^n;r)[x]/(xe - pu ; x^s),$ 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.