Prime Ideal is Irreducible in a Commutative Ring
Problem 173
Let $R$ be a commutative ring. An ideal $I$ of $R$ is said to be irreducible if it cannot be written as an intersection of two ideals of $R$ which are strictly larger than $I$.
Prove that if $\frakp$ is a prime ideal of the commutative ring $R$, then $\frakp$ is irreducible.
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Proof.
Suppose that the ideal $\frakp$ is reducible. Then there exist ideals $I_1$ and $I_2$ such that
\[\frakp=I_1 \cap I_2, \text{ and } \frakp \subsetneq I_1, \frakp \subsetneq I_2.\]
Since $I_1, I_2$ are strictly larger than $\frakp$, there exists $a\in I_1\setminus \frakp$ and $b\in I_2 \setminus \frakp$.
Then the product $ab\in I_1$ since $a$ is in the ideal $I_1$. Also $ab \in I_2$ since $b$ is in the ideal $I_2$.
Therefore $ab\in I_1 \cap I_2=\frakp$.
Since $\frakp$ is a prime ideal and $ab \in \frakp$, either $a\in \frakp$ or $b \in \frakp$ but this contradicts with the choice of elements $a$ and $b$.
Hence $\frakp$ is irreducible.
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