Composing the generators of an ideal with a monic polynomial

Question

Suppose $I = (f_{1}, \ldots, f_{n})$ is an ideal in a polynomial ring $R[x]$, where $R$ is a commutative ring with identity, minimally generated by $n$ elements. Let $p \in R[x]$ be a monic nonconstant polynomial and define the ideal $I(p) = (f_{1}(p), \ldots, f_{n}(p))$. Is it possible that $I(p)$ has a generating set with fewer than $n$ polynomials?