Incorrect proof- why?

Question

I know that this proof for $1+1≠0$ is incorrect- can someone explain where the logic is incorrect?

Proof by contradiction:

Assume $1+1=0.$ Then:

$$2=0$$ $$2\cdot\frac12=0\cdot\frac12$$ $$1=0$$ ...which is a contradiction.

Answer 1

The question could probably use some clarification. You are making two assumptions.

The first, explicit, that $1+1$ equals $0$ (which is correct e.g. in $\mathbb{F}_2$).

The second, implicit, that $1+1=2$ has a multiplicative inverse (by writing it as $1/2$).

The two assumptions together indeed imply that $1=0$, which is no contradiction: it just means you are in the Zero Ring.