Is $\mathbb{Z}\oplus \mathbb{Z}_3\cong\mathbb{Z}$ as modules

Question

I'm studying module theory, and I'm trying to prove whether $\mathbb{Z}\oplus \mathbb{Z}_3\cong\mathbb{Z}$ as modules. If i construct the map $\mu :\mathbb{Z}\oplus \mathbb{Z}_3\longrightarrow \mathbb{Z}$ sending $(a,[b])\mapsto 3a+b$, where we associate $[0]$ with $0$, $[1]$ with $1$ and $[2]$ with $2$, I would expect $\mu$ to be an isomorphism of modules. Or is there a problem with my use of residual classes? I feel a bit like this map is not well defined, but I can't put my finger on why.