In a set of old lecture notes, I came across this corollary:
Corollary: Any maximal ideal is a prime ideal.
While I do not have the proof for this, what immediately comes to mind is whether this corollary is predicated on the fact that...
Theorem: For any commutative ring $R$ and ideal $A$, $R/A$ is an integral domain iff $A$ is a prime ideal
Theorem: For any commutative ring $R$ and ideal $A$, $R/A$ is an field iff $A$ is a maximal ideal
Recall: Any finite integral domain is a field.
So, from the two theorem above, we can reconcile the fact that a maximal ideal is a prime ideal (and vice versa) on the supposition that the integral domain in question is finite.
Or is there another proof for the claims in my notes?