Which is a subring? Which is an ideal?

Question

We are having ring $\mathbb{Z}[\sqrt{-6}]$. Which of the sets is subrings of $\mathbb{Z}[\sqrt{-6}]$ and which are ideals?

1. $\mathbb{Z}+5\mathbb{Z}[\sqrt{-6}]$
2. $5\mathbb{Z}+\mathbb{Z}[\sqrt{-6}]$
3. $2\mathbb{Z}+3\mathbb{Z}[\sqrt{-6}]$

I know that if the ring is ideal of some set, then it is automatically a subring of this set, but that works only one way. If we have a subring, we can't say that it is an ideal.

To prove that a subring is an ideal we must present for example element from first set as $a+5b\sqrt{-6}$, where $a,b \in \mathbb{Z}$ and multiply it on similar representation of $\mathbb{Z}[\sqrt{-6}]$: it will be $a+b\sqrt{-6}$. And then if result $\in \mathbb{Z}+5\mathbb{Z}[\sqrt{-6}]$ we can say that it is an ideal.

Am I doing right way? And can you help me with proving that set is subring? (What should we say, to prove that set is subring).

Thanks for any help!

Answer 1

To prove that a set $I\subset R$ is an ideal, you must take an element $x$ from $I$ and an element $r$ from $R$ and prove that $x\cdot r$ is still an element of $I$. For example, for your first example, take $a+5b\sqrt{-6}$ from the "ideal" and $c+d\sqrt{-6}$ from the ring. Multiply the two to get $(ac - 30bd) + (5bc + ad)\sqrt{-6}$. Is this an element of $\mathbb Z + 5\mathbb Z [\sqrt{-6}]$ or can you find a counterexample where this does not hold?