When do quadratic polynomials generate the same ideal?

[Mr Davis 97]

Homework Statement

When do two quadratic polynomials in $\mathbb{Z}_3 [x]$ generate the same ideal?

Homework Equations

The Attempt at a Solution

I feel like they generate the same ideal only when they have the same coefficients, but am not sure how to show this.

 

[fresh_42]

It's not necessary for your question, as you only asked about principal ideals. Nevertheless, is $\mathbb{Z}_3[x]$ a principle ideal domain?
Now to approach your question. As always, write down what is given, namely two ideals $I=\langle p(x) \rangle$ and $J=\langle q(x)\rangle$ with polynomials $p(x),q(x)\in \mathbb{Z}_3[x]$ and $I=J$. The latter means $p(x) \in J$ and $q(x) \in I$. What can you conclude from this?