Equivalency of three statements about localization of a ring

Question

I want to show that these are equivalent for any ring $R$:

1. For every prime ideal $P\subset R $, $R_P $ is a domain, where $R_P=\{r/s : r\in R , s\in R-P \}$.

2. For every maximal ideal $m\subset R$, $R_m$ is a domain.

3. If $ab=0$, then $\mathrm{Ann}(a)+\mathrm{Ann}(b)=R$, where $\mathrm{Ann}(a)=\{x\in R: xa=0\}$.

I tried to derive $3$ from $2$, but I could just prove $\mathrm{Ann}(a)+m=R $, when $m$ is a maximal ideal contains $\mathrm{Ann}(b)$.

If ring is commutative, $1$ gives $2$ trivially.

Is there any hint? Thanks