On an extension of the Newton polygon test for polynomial reducibility

Let R be a commutative local principal ideal ring which is not integral, f a polynomial in R[x] such that f (0) ̸ = 0 and N (f ) its Newton polygon.If N (f ) contains r sides of different slopes, we show that f has at least r different pure factors in R[x].This generalizes the Newton polygon method over a ring which is not integral.