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This question is Exercise 1.2.20 in the book: Winfried Bruns, H. Jürgen Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1998.

Let $k$ be a field and $R=k[[X]][Y]$. Deduce that $X, Y$ and $1-XY$ are maximal $R$-regular sequences.

I can not get why $1-XY$ is a maximal $R$-regular sequence. How to check the maximal condition?

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Since $R$ is an integral domain, it follows that $1-XY$ is a non-zerodivisor in $R$. From this topic we know that $(1-XY)$ is a maximal ideal in $R$, so $1-XY$ can't be extended to an $R$-sequence since $(1-XY,a)=R$ for any $a\in R$, $a\notin(1-XY)$.

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