The function is $L_{[a]}([b]) = [a][b] = [ab]$, where $[a],[b] \in \mathbb{Z}_n$ and $[a]$ is not a unit.
I know that it is not surjective, since $[a]$ is not a unit so no element will give $[1]$, but I am lost for injectivity.
Basically I think I have to show that zero divisors are not unique, but I do not know how to show this for the general case $\mathbb{Z}_n$.