Are there examples of nontrivial subrings of $\mathbb{R}$ that do not contain $1$? If not, how can we prove they don't exist? The definition of "ring" here is really "rng"; rings do not have to contain $1$.
If $m \in \mathbb{Z}$, then $\mathbb{Z}[\sqrt{m}] = \{ a + b \sqrt{m} : a, b \in \mathbb{Z} \}$ is a subring of $\mathbb{R}$, but it contains $1$. Similarly, $\mathbb{Z}$ and $\mathbb{Q}$ are subrings, but they also contain $1$.
EDIT: there are quite a lot of counterexamples ($\mathbb{Z}a$ for $a \in \mathbb{Z}_{\geq 2}$, e.g. the even integers $\mathbb{Z}2$). I am still interested in seeing other counterexamples. Is there a way to classify all the counterexamples?