Let $A$ be a commutative ring with unity. Prove: Proof every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.)
In the question before this I proved that $J$ is a prime ideal iff $A/J$ is an integral domain. Now, I have what I think is a "pseudoproof" and as such, am not satisfied.
$\rightarrow$ Let $J$ denote an arbitrary maximal ideal of $A$. Since $J$ is a maximal ideal of $A$, $A/J$ is a field. Because every field is an integral domain, $A/J$ is an integral domain. Since $A/J$ is an integral domain, $J$ is a prime ideal. Thus, every maximal ideal $J$ of $A$ is a prime ideal.
Is there another way to prove this directly?