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No, in the sense that this statement is false in a ring without unique factorization. For example, the polynomial $x^2 y$ is irreducible in $k[x^2, xy, x^2 y]$, and $x^2 y^2 \in (x^2 y)$ but $xy \not \in (x^2 y)$.

Here is a counterexample where the ring is even integrally closed. 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.)

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No, in the sense that this statement is false in a ring without unique factorization. For example, the polynomial $x^2 y$ is irreducible in $k[x^2, xy, y^2]$x^2 y]$, and$x^2 y^2 \in (x^2 y)$but$xy \not \in (x^2 y)$. Here is a counterexample where the ring is even integrally closed. 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.) 2 added 274 characters in body No, in the sense that this statement is false in a ring without unique factorization. For example, the polynomial$x^2 y$is irreducible in$k[x^2, xy, y^2]$. Moreover, and$x^2 y^2 \in (x^2 y)$but$xy \not \in (x^2 y)$. Here is a counterexample where the ring is even integrally closed. 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.)

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