Sperner’s theorem for non-free modules over finite chain rings

We prove Sperner-type theorems for the partially ordered set \(\mathcal {P}_M\) ${\mathcal{P}}_M$ of all submodules of a non-free finitely generated module \({}_RM\) ${}_{R}M$ over a finite chain ring R. We demonstrate that the partially ordered set \(\mathcal {P}_M\) ${\mathcal{P}}_M$ is not necessarily of Sperner type and solve the problem for modules of shape \(2^11^n\) $2^1{1}^n$ . This result is further generalized for modules of shape \(m^11^n\) $m^1{1}^n$ over a chain ring of length m.