Let A and B be ideals of a ring and C a prime ideal. Prove if the intersection of A and B is a subset of C then either A or B is a subset of C

Question

Claim: Let $A$ and $B$ be ideals of a commutative ring $R$ and $C$ a prime ideal of $R$. Suppose that the intersection of $A$ and $B$ is a subset of $C$. Prove either $A$ or $B$ is a subset of $C$.

My proof strategy so far is to show $a$ $\in$ $A$ and $b$ $\in$ $B$ implies $ab$ is in both $A$ and $B$ since both are ideals thus $ab$ $\in$ $A \cap B$. Then $ab$ is also in C since the intersection of $A$ and $B$ is a subset of $C$. And since $C$ is prime we must have either $a$ $\in$ $C$ or $b$ $\in$ $C$ whence either $A$ or $B$ is a subset of $C$.

Would my proof be suffice or am I missing anything? Thanks.

Answer 1

The problem is that you've only shown one element of $A$ or $B$ is contained in $C$. To prove the full statement, you assume for contradiction that there exist $a\in A-C, b\in B-C$, and then your argument gives a contradiction.

Answer 2

Suppose $A\not\subseteq C$ and take $a\in A$, $a\notin C$.

If $b\in B$, then $ab\in C$, so…

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Sorry, but your proof is missing the main point, which is to show that, if one of the ideals is not contained in $C$, then the other one is.

You just proved that, given $a\in A$ and $b\in B$, then one of them is in $C$, which is not sufficient.