Is it true that $\mathbb{Z}[\sqrt{-3}] + \mathbb{Z}[\sqrt{-5}]$ not a ring?

Question

Question: Is it true that $$\mathbb{Z}[\sqrt{-3}] + \mathbb{Z}[\sqrt{-5}]$$ not a ring?

I think it is not because it is not closed under multiplication. More precisely, if $a_1+b_1\sqrt{-3}, a_2+b_2\sqrt{-3}\in \mathbb{Z}[\sqrt{-3}]$ and $c_1+d_1\sqrt{-5},c_2+d_2\sqrt{-5}\in \mathbb{Z}[\sqrt{-5}],$ then $$(a_1+b_1\sqrt{-3} + c_1+d_1\sqrt{-5}) \cdot (a_2+b_2\sqrt{-3} + c_2+d_2\sqrt{-5})$$ is not an element of $\mathbb{Z}[\sqrt{-3}] + \mathbb{Z}[\sqrt{-5}].$

Answer 1

You don't even have to be so general about it. A single counterexample is enough (and often easier to actually prove).

For instance, $\sqrt{-3}$ and $\sqrt{-5}$ are both in your set, but their product (whether that's $-\sqrt{15}$ or $\sqrt{15}$) isn't.

However, $\Bbb Z[\sqrt{-3}]+\Bbb Z[\sqrt{-5}]+\Bbb Z[\sqrt{15}]$ is a ring.