Example of an infinite ring with an odd number of units?

Question

I have my abstract algebra exam online this thursday and was hoping to get some confirmation on some of these short questions.

Is there an example of an infinite ring with an odd number of units?

Every non-field example I can think of simply has two: $\{\pm 1\}$

Is there a field of characterstic $7$ which has an element $x$ such that $x+x+x=0$?

I think the answer must be no. The only fields I have in mind are $\frac{\mathbb{Z}}{7\mathbb{Z}}$ and $\frac{\mathbb{Z}[x]}{7\mathbb{Z}[X]}$(rational functions).

Answer 1

$\mathbb{Z}_2$ has only one unit, $1$ since $-1=1$ here. Can you use this to create an infinite ring that would still have only $1$ as its unit?

If the characteristic is $7$, then $7x=x + x + ... + x =0$ for all $x$. This means the order of every element under addition must divide 7, so either order 1 or 7. You could of course let $x=0$ in $\mathbb{Z}_7$, but I assume they don't want a zero solution.

Answer 2

For the second question, more generally we have

There is no ring of characteristic $7$ which has a nonzero element $x$ such that $x+x+x=0$.

Indeed, $7x=0$ and $3x=0$ imply $x=\gcd(7,3)x=0$.