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

中文翻译：

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有限链环上非自由模的斯佩纳定理

我们证明了有限链环R上非自由有限生成模块\({}_RM\)的所有子模块的偏序集\(\mathcal {P}_M\ )的 Sperner 型定理。我们证明偏序集\(\mathcal {P}_M\)不一定是 Sperner 类型，并解决形状为\(2^11^n\)的模的问题。这个结果进一步推广到长度为m的链环上形状为\(m^11^n\)的模块。