Let $P$ be a prime ideal in a commutative ring $R$ and let $S=R-P$ ,i.e. $S$ is the complement of $P$ in $R$. Then, justify with reason which of the following(s) are correct:
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$S$ is closed under addition.
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$S$ is closed under multiplication.
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$S$ is closed under both addition and multiplication.
The following argument provides a partial answer:
Let $P=3 \mathbb Z$ and $R= \mathbb Z$ the $2,4$ in $S$ but $2+4$ in $P$, so option 1. and 3. are incorrect.
But I don't know about 2.
Best Answer