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YCor
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Ideal in ring of power series

Let $K$ be a field of characteristic $p$ and $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring over $K$ such that $n, p \geq 3$.

Consider the ideal $I$ defined by \begin{equation*} I \colon= (X_1^{p},X_2^{p^2},X_3^{p^3},\ldots,X_n^{p^n}). \end{equation*} Suppose that $\alpha, \beta \in A_n$ be two different prime elements such that $\alpha \notin I$ and $\beta \notin I$.

Q. Does it always hold that the multiplication $\alpha \beta \notin I$?

Pierre
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