Let's take $p$ a prime number and define $\epsilon^2 = 0$.
If I understand well:
$$(\mathbb{Z}/p\mathbb{Z})[\epsilon] \simeq \mathbb{F}_p[\epsilon] \simeq \mathbb{F}_p[X]/(X^2\mathbb{F}_p)$$
But then, do we have:
$$(\mathbb{Z}/p\mathbb{Z})[\epsilon] \simeq \mathbb{Z}/p^2\mathbb{Z}$$
?
There is a simple evaluation morphism that sends $\epsilon$ to $p$. Are they the same? I am confused because I'm reading a cryptic cryptography article where both are used but it is not stated they are the same.
There is no such isomorphism: on the left, every element times $p$ is zero, while on the right, not so much.
Technically, the exponent of the underlying abelian groups are $p$ and $p^2$, so there is no isomorphism of rings, as there is no isomorphism of abelian groups.