= Maximal annihilator of a module element is prime
Over a commutative unital <ring>, an ideal maximal among annihilators of nonzero elements of a <module> is a <prime ideal>. Each such ideal is the <annihilator of a module> given by the cyclic submodule generated by that element. If $P=\operatorname{Ann}(m)$ and $ab\in P$ with $b\notin P$, then $bm\ne0$ and $P\subseteq\operatorname{Ann}(bm)$. Maximality makes these annihilators equal, so $a\in P$. A <Noetherian ring> ensures that a nonzero module has a maximal element annihilator by the <ascending chain condition> on ideals.
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