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 $\beta \notin I$.