Subgroup of $\mathbb{Z} \oplus \mathbb{Z}$ that is not a subring? I just had a test, and one of the questions was to show that there is at least one subgroup of $\mathbb{Z} \oplus \mathbb{Z}$ that is not a subring.
I couldn't think of one and still can't, so I cheated and said $\{ (2k,2k) \colon  \ k\in \mathbb{Z} \}$ is a subgroup but not a (unitary) subring... In this class we don't require a ring to have identity so it is wrong obviously.
Anyways can you please give me an example?
Thank you

Hint: Observe that $(1,2)·(1,2) = (1,4)$.
