Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal.
Hint: Give a counter-example
This is my rough proof to this question. I was wondering if anybody can look over it and see if I made a mistake or if there is a simpler way of doing this problem. So lets begin:
Ideal generated by $4$ in $\mathbb Z_{12}$ is given by: $I=\{0,4,8\}$ since, $2\times2= 4\in I$. But $2\notin I\Rightarrow I$ is not prime ideal.
I want thank you ahead of time for taking the time to look at this problem.