I'm fairly new to ideals in my algebra course, and I understand the basics of ideals, such that I is an ideal iff (I,+) is a subgroup of (R,+) (using normal subgroup tests) and for all $r \in R$ and $a \in I$ then $ ar,ra \in I$. As a question, I've been given a set and asked to list the ideals.
The set given is $\mathbb{Z}$/$20\mathbb{Z}$. My first thought was to try and come up with an example or two; I came up with $I=$ { ${n\in \mathbb{Z}}||n|\geq0 $}; this seems like an ideal as it has the $0$ element, contains the additive property, and for any $r\in R$, $r n \in \mathbb{Z}/20\mathbb{Z}$.
However this seems like a very plain and obvious example that I just 'made work.' I'm not sure if there's a method or any way I should be going about finding these ideals. Any help would be great.
Edit: I've found that all subgroups of a cyclic group are also cyclic of different sizes. The sizes are divisors of the size of the main group. Therefore in a cyclic group of size 20 there are 6 subgroups:
the original group,
{$0,2,4,6,8,10,12,14,16,18$} $=g^{10}$,
{$0,4,8,12,16$} $=g^5$,
{$0,5,10,15$} $=g^4$,
{$0,20$} $=g^2$,
{$0$} = $g^1$
First you should remember that $\mathbb{Z}/20\mathbb{Z}$ is a set of equivalence classes; usually we choose the representatives $\{0,1,2,3,4,5,...,19\}$. So when you say all members from here $\ge 0$ you're essentially taking $\mathbb{Z}/20\mathbb{Z}$ to be an ideal of itself. And it turns out (rather obviously) that it's an ideal if you're not requiring that an ideal has to be a proper subset of the ring.
You can quickly check if $\{0\}$ is an ideal (it is, called the trivial ideal). For the other groups, to get familiar with ideals you might want to see what happens when you're testing whether or not $ra \in I$. Can you take anything from $\mathbb{Z}/20\mathbb{Z}$ and multiply it by anything in $g^{5}$ to get something not in $g^{5}$? These are the sorts of questions you want to ask.