Polynomials taking integer values on primes in a function field

Let $\mathbb{F}_q[x]$ be the ring of polynomials over a finite field $\mathbb{F}_q$ and $\mathbb{F}_q(x)$ its quotient field. $\mathbb{P}$ set primes in $\mathbb{F}_q[x]$, let $\mathcal{I}$ all $f$ for which $f(\mathbb{P})\subseteq\mathbb{F}_q[x]$. The existence basis is established using notion characteristic ideal; this shows that free $\mathbb{F}_q[x]$-module. Through localization, explicit shapes certain bases localization are derived, well-known procedure described as to how obtain forms some $\mathcal{I}$.