Exercise 111.12.8. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $x, y \in \mathfrak m$ be a regular sequence of length $2$. For any $n \geq 2$ show that there do not exist $a, b \in R$ with

Suggestion: First try for $n = 2$ to see how to argue. Remark: There is a vast generalization of this result called the monomial conjecture.

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