What are the Ideals of $Z/5Z$. Now $Z/5Z$ is going to form a quotient ring .Will this ring have an ideal? My intuition is the ideal will be 0 (the only ideal)?.
To find the ideal structure of a ring $\mathbb Z/n\mathbb Z$ you can consider the ideal structure of $\mathbb Z$ which contains $n\mathbb Z$. for example 





So the ideals of $\mathbb Z/5\mathbb Z$ are $\mathbb Z/5\mathbb Z$ and 0
The only ideals in $\Bbb Z/n\Bbb Z$ are those of the form $\Bbb Z/m\Bbb Z$, where $m|n$.

Since $5$ is prime, there are only $0$ and $\Bbb Z/5\Bbb Z$, which, incidentally, means it's a field.