I'm asked to prove that the ring of integers $\Bbb Z$ admits no other subring than itself.
I'm no too sure about how to prove it. I started using the minimality argument, but I found a counter-example - which must be wrong but I would like to know why. Actually, I thought to the set of all even number of $\Bbb Z$. This set is not empty, close under addition and multiplication. Where is my mistake in this example, and what could be a rigorous proof of this statement ?