A basic question about finding ideals of rings and proving that these are all the ideals I am a student taking a "discrete maths" course. Teacher seems to jump from one subject to another rapidly and this time he covered ring theory, Z/nZ, and polynomial rings.
It is hard for me to understand anything in his class, and so the reports he gives become very hard.
I did my best to find answers using google, but I just couldn't find it.
Specifically he asked us to find all ideals of Z/6Z, and prove that these are in fact all of them.
He also asked us to find all ideals of F[X]/(X^3-1) where F stands for Z/2Z.
I understand the idea behind ideals, like I can see why {0,3} is ideal of Z/6Z, but how do I find ALL the ideals?
And regarding polynomials, is there some kind of a mapping between polynomials and Z/nZ?
Because otherwise I have no idea how to find ideals of polynomials.

They're both products of fields: $\mathbb{Z}/6\cong\mathbb{Z}/2\times\mathbb{Z}/3$ and $F[x]/(x^3-1)\cong GF(2)\times GF(2^2)$ (if you've covered this kind of thing) which makes it easy to see the idea structure.
