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David Roberts ♦
David Roberts ♦
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If $P=I\cap J$ is prime, and if $P$ is not equal to $I$ or $J$, then choose $i\in I-J$$i\in I\setminus J$ and $j\in J-I$. (The minus sign means set-theoretic complement$j\in J\setminus I$.) Then Then $ij\in P$, so $i\in P$ or $j\in P$. Contradiction either way. Thus $P=I$ or $P=J$.

If $P=I\cap J$ is prime, and if $P$ is not equal to $I$ or $J$, then choose $i\in I-J$ and $j\in J-I$. (The minus sign means set-theoretic complement.) Then $ij\in P$, so $i\in P$ or $j\in P$. Contradiction either way. Thus $P=I$ or $P=J$.

If $P=I\cap J$ is prime, and if $P$ is not equal to $I$ or $J$, then choose $i\in I\setminus J$ and $j\in J\setminus I$. Then $ij\in P$, so $i\in P$ or $j\in P$. Contradiction either way. Thus $P=I$ or $P=J$.

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Steven Landsburg
Steven Landsburg
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If $P=I\cap J$ is prime, and if $P$ is not equal to $I$ or $J$, then choose $i\in I-J$ and $j\in J-I$. (The minus sign means set-theoretic complement.) Then $ij\in P$, so $i\in P$ or $j\in P$. Contradiction either way. Thus $P=I$ or $P=J$.