Why isn't $\displaystyle Z_2 = \{0, 1\}$ a subring of the integer set $\displaystyle Z$?
Ah! But remember it is not really $\displaystyle \{ 0,1\}$ it is really $\displaystyle \{ [0]_2,[1]_2\}$.
Where, $\displaystyle [0]_2 = \{ 0, \pm 2, \pm 4, ... \}$ and $\displaystyle [1]_2 = \{ \pm 1, \pm 3, \pm 5, ... \}$.
Therefore, $\displaystyle \mathbb{Z}_2$ is a group modulo $\displaystyle 2$ under addition.
I donīt think it was necessary to lower my reputation for my comment; obviously we are all talking about rings with unity, so my comment about the subrings of $\displaystyle \mathbb{Z}$ is correct in that context.
Bringing to the table the issue of pseudo-rings (or rngs) had nothing to do with the question, must insist...