I'm looking for an example for a left R-module that doesn't have the lifting property, From theorems I read, $Z/2Z$ as a $Z$ module should be an example since it' not a direct sum (there isn't a sub-module $K$ of $Z$ such that the direct sum of $Z/2Z$ and $K$ is $Z$).
But I don't understand why $Z/2Z$ doesn't have the lifting property, can someone explain why?