I've got two questions concerning divisible modules that are defined as follows: Let $M$ be a $R$-module. $M$ is called divisible iff for all $r\neq 0$ the map $\phi_r :M \to M $ $m\mapsto rm$ is surjective.
Questions: 1) Is the map $\phi_r$ also injective?
[ANSWERED in the comments: Consider the $\mathbb Z$-module $\mathbb{Q}/\mathbb{Z}$.]
2) Is every injective $R$-module divisible? (I found this to be true over $R$ being PID or integers, but does it hold also for general $R$?)
Thank you for your help!