What are the Ideals of $Z/5Z$.

Question

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)?.

Answer 1

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

Answer 2

For a prime number $p$, $\mathbb{\mathbb{Z}}/p\mathbb{Z}$ is field, and in a field only ideals are itself ($\mathbb{\mathbb{Z}}/p\mathbb{Z}$) and $\{0\}$.

Answer 3

It is a field. It's only ideals are 0 and itself.

Answer 4

There are two ideals that every ring(commutative, with unit) has: the ideal generated by $0$ and the ideal generated by $1$.

If $k$ is an arbitrary field, then $k$ has exactly two ideals, namely $(0)$ and $(1)=k$, since every nonzero element is a unit and therefore we have for every ideal $(x)$ with $x \neq 0$ that $1 \in (x)$ and therefore $(x)=k$.

Since $\mathbb{Z}/5\mathbb{Z}$ is a field, there are only those two ideals.