Can you tell me what's wrong with my proof? Thanks.
Claim: If $R$ is a Noetherian ring and $I$ is an irreducible ideal in $R$ then $I$ is prime
Proof: Let $xy \in I$. We want to show that either $x\in I$ or $y \in I$. By contradiction assume neither $x\in I$ nor $y \in I$. Then $I + \langle x \rangle$ and $I + \langle y \rangle$ are two ideals properly containing $I$ and $I + \langle x \rangle \cap I + \langle y \rangle = I + \langle xy \rangle = I$, i.e. $I$ is reducible which is a contradiction to our assumption.