Let $U = \{1,0,c\}$ be a ring with three elements ($1$ is the unity). Which
statements are true?
\begin{align*}
&I.\,\,\,\,\,\,\,\,1 + 1 + 1 = 0 \\
&II.\,\,\,\,\, 1 + 1 = c \\
&III.\,\,c^2 = 1
\end{align*}
The solution is I II and III are all true. For III, the solution says by multiplicative table, $c^2$ has to be $1$. I know $c*1=c$ and $c*c=c$ is a bit weird. But I don't think it contradicts the definition of Ring, which just requires closure and associative for multiply.
Note that $c = 1 + 1$ must hold, so
$$c^2 = 1^2 + 1 + 1 + 1^2 = 1 + 1 + 1 + 1 = 0 + 1$$
To see the very first statement, note that if $1 + 1 = 1$, then $1 = 0$; if $1 + 1 = 0$, then this contradicts that the additive group must have order $3$.