Property about $a+b=0$

Question

We know well that $$ab=0\iff a=0 \vee b=0.$$ Is there any property that says about $a+b=0$? What about $$a+b=0 \stackrel{?}\iff a=0\wedge b=0.$$ Note that I do not deny that there are many solutions of it. I was studying about the metric for fun and the proof of 1) condition on taxicab metric is a metric made me think about this property.

Answer 1

If you know $a,b\ge 0$, it is true. For instance $$a^2+b^2=0\implies a^2=b^2=0\iff a=b=0,$$ $$\lvert a\rvert+\lvert b \rvert=0\implies \lvert a\rvert=\lvert b \rvert=0\iff a=b=0.$$

Answer 2

$$ a=0 \land b=0 \Rightarrow a+b=0 $$ is true, but $$ a+b=0 \Rightarrow a=0 \land b=0 $$ is false (in a ring).