Homomorphisms between $\mathbb {F_p}$ and $\mathbb F_{p^2} \bigoplus \mathbb F_p$.

Question

I’m looking for a good method to count number of ring homomorphisms between 2 rings $\mathbb {F_p}$ and $\mathbb F_{p^2} \bigoplus \mathbb F_p$. I don’t know how I can find ALL subfields isomorphic to $\mathbb F_p$ in the direct sum and how I can guarantee that no elements of the field have a zero divisor image. Thank you in advance!

Answer 1

Let $A$ be a ring with $1$. Ring morphisms $\mathbb{F}_p\to A$ are in 1-1 correspondence with ring morphisms $f: \mathbb{Z}\to\ A$ such that $\ker(f)\supset p\mathbb{Z}$.

There is exactly one ring morphism $\theta:\mathbb{Z}\to A$, and it is defined by $\theta (m)=m\cdot 1_A$.

Now $p\in\ker(\theta)$ if and only if $p\cdot 1_A=0$, meaning that $A$ is either trivial or have characteristic $p$.

Conclusion. for any ring with $1$, there is no ring morphism $\mathbb{F}_p\to A$ if $A$ is not trivial has characteristic $\neq p$, and there exists exactly one ring morphism $\mathbb{F}_p\to A$ if $A$ has characteristic $p$.

In particular, the second possibility occurs in your situation.