# 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 $$\alpha \beta \notin I$$?

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 $$I$$.