Generalizing Sperner's lemma to a free module over a special principal ideal ring (Q1941831)

From the author's abstract: ``Sperner's lemma states that if \(\mathcal{A}\) is an anti-chain from the power set of an \(n\)-element set, then \(|\mathcal{A}| \leq \binom{n}{\lfloor \frac{n}{2}\rfloor}\). Rota and Harper provide the following \(q\)-analogue to a number of classical generalizations of Sperner's lemma: If \(\mathcal{A}\) is an \(l\)-chain-free family of subspaces of a finite vector space \(\mathbb{F}^n_q\), then \(\sum_{A\in \mathcal{A}}\frac{1}{\binom{n}{\dim(\mathcal A)}_q} \leq l\) and \(|\mathcal{A}|\) is bounded by the sum of the \(l\) largest Gaussian coefficients \(\binom{n}{k}_{q}\). In this work, the original Sperner's lemma as well as Rota and Harper's result are extended to multiple generalizations in the setting of a finitely-generated free module over a finite special principal ideal ring.'' Here, the \textit{finite special principal ideal ring} \(R\) is a finite principal ideal ring, which has only one prime ideal \(M\) and \(M^i=0\) for some \(i>0\). This work mainly deals with the nontrivial case when \(i>1\); when \(i=0\), the ring \(R\) is indeed a finite field. This work is mainly based on the previous paper by the author [Rend. Circ. Mat. Palermo (2) 51, No. 1, 5--50 (2002; Zbl 1183.13029)]. The technical tools are developed as the Appendix of this paper. After all these preparations, the proof for the main results of this work is almost standard and straightforward.