Ideal in ring of power series

Question

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 $\alpha \beta \notin I$?

Answer 1

The answer is "no". Take $n=3$ and an arbitrary prime $p\ge3$. Let $\alpha=X_3^{p^3}+X_1X_2^{p^2-1}$ and $\beta=X_3^{p^3}+X_1^{p-1}X_2$. Both are irreducible elements, not belonging to $I$, but their product is in $I$.