Let $\Bbb Z/p$ be the finite field with $p$ elements.
Consider $\Bbb Z/p[X]$, the ring of polynomials with coefficients in $\Bbb Z/p$.
Consider also the ring $P(\Bbb Z/p)$ of all polynomial functions on $\Bbb Z/p$.
Let $\varphi$ be the morphism $\Bbb Z/p[X]\to P(\Bbb Z/p)$, linking to each polynomial its polynomial function.
Is it correct that the kernel of $\varphi$ is the ideal $(X^p-X)$?