Let $R$ be the ring $\mathbb{Z}/n\mathbb{Z}$ with $n \in \mathbb{N}$ composite and consider $R$ as a right module over itself.
I'd like to determine the projective submodules of $R$ if there are any.
First $R$ is free over itself, $\{1+n\mathbb{Z}\}$ being a base.
Now as an additive group $R$ is the direct product of its Sylow subgroups.
My doubts begin here: is it correct to infer that as a right $R$-module $R$ is the direct sum of its Sylow subgroups seen as right $R$-submodules?
If it were so then each Sylow subgroup is a projective $R$-submodule being a direct addend of a free $R$-module.
Clearly none of the former is free since it contains torsion elements.
Next question is: is there any other projective submodule?
If the answer is no then I just proved that the ring $\mathbb{Z}/n\mathbb{Z}$ is hereditary if and only if $n$ is square-free, right?
Any suggestion or critique is widely appreciated.
Best Answer
Modules over $R=\mathbb{Z}/n\mathbb{Z}$ are exactly the same as modules over $\mathbb{Z}$ on which $n$ acts trivially. So an abelian group on which $n$ acts trivially splits as a direct sum of abelian groups if and only if it splits as a direct sum of $R$-modules.