Let $R$ be infinite commutative ring with unity
and $I$ be finite prime ideal of $R$
I want to know if $I$ must be trivial?
I can prove it if we further assume $R$ is integral domain
Proof:
Since $I$ is nontrivial,
$\exists a \in I$ such that $a \ne 0$
Since R is infinite,
we can pick distinct $r_1$, $r_2, \dots \in R$
Now $\{r_i a \mid i \in \mathbb{N}\} \subseteq I$
Since $I$ is finite, $\{r_i a \mid i \in \mathbb{N}\}$ is also finite
So $\exists r_i, r_j \in R$ with $r_i \ne r_j$
such that $r_i a = r_j a$
Hence $(r_i - r_j) a = 0$
Since R is integral domain and $a \ne 0$,
$r_i = r_j$
Contradiction, $I$ is trivial
I want to know if this is true in the general case?