Let $A$ be an integral domain, $M$ an $A$-module, and $m\in M$. Now for all maximal ideals $\mathfrak{m}$ there exists an $n\notin \mathfrak{m}$ such that $nm=0$. Why does this mean that $m=0$?
2026-03-28 01:14:57.1774660497
Relationship between modules and maximal ideals of a commutative ring
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Hint. Show that the ideal generated by $n\notin\mathfrak m$, when $\mathfrak m$ runs over the maximal ideals of $A$, is the whole ring.