Theorem(Nilradical lemma). Let $K$ be a field of characteristic $p > 2$ and $A_{\infty} \colon= K[[X_1,\ldots,X_{\infty}]]$ be an infinitely-many-variable formal power series ring over $K$. We shall define the closed ideal $I$ of $A_{\infty}$ as follows$\colon$
\begin{equation*}
I \colon= (X_1^p,...
let $$\Pi\binom{n}{k}:=\mathrm{card}\left( \left\lbrace \lbrace \Pi_1^n\,\cdots\,\Pi_k^n\rbrace\,|\,0\leq \pi_{r,c}\in\sum_{i=1}^k\Pi_i^n\ni\pi_{r,c}\leq 1\right\rbrace\right)$$ be the number of sets with $k\leq n$ distinct $n\times n$ permutation matrices without common non-zero entry.
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let $$\Pi\binom{n}{k}:=\mathrm{card}\left( \left\lbrace \lbrace \Pi_1^n\,\cdots\,\Pi_k^n\rbrace\,|\,0\leq \pi_{r,c}\in\sum_{i=1}^k\Pi_i^n\ni\pi_{r,c}\leq 1\right\rbrace\right)$$ be the number of sets with $k\leq n$ distinct $n\times n$ permutation matrices without common non-zero entry.
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