No, in the sense that this statement is false in a ring without unique factorization. For example, the element $2 + \sqrt{-5}$ is irreducible in $\mathbb{Z}[\sqrt{-5}]$, and $9 \in (2 + \sqrt{-5})$ but $3 \not \in (2 + \sqrt{-5})$.

(The lesson here is that irreducibility is not a useful idea in non-UFDs.)