On the integral ideals of R[X] when R is a special principal ideal ring

Let $$(R, \pi R, k,e)$$ be a commutative special principal ideal ring (SPIR), where R is its maximal ideal, k its residual field and e the index of nilpotency of $$\pi$$ . An ideal I of R[X] is called an integral ideal if it contains a monic polynomial. In this paper, we show that if R is a SPIR, then an ideal I in R[X] is integral if and only if $${\overline{I}} \ne {{\overline{0}}}$$ in k[X]. Furthermore, the lowest degree of monic polynomials in I is exactly the lowest degree of nonzero polynomials in $${\overline{I}}$$ .