Lemma 10.66.7. Let R be a ring. Let M be an R-module. The union \bigcup _{\mathfrak q \in \text{WeakAss}(M)} \mathfrak q is the set of elements of R which are zerodivisors on M.
Proof. Suppose f \in \mathfrak q \in \text{WeakAss}(M). Then there exists an element m \in M such that \mathfrak q is minimal over I = \{ x \in R \mid xm = 0\} . Hence there exists a g \in R, g \not\in \mathfrak q and n > 0 such that f^ ngm = 0. Note that gm \not= 0 as g \not\in I. If we take n minimal as above, then f (f^{n - 1}gm) = 0 and f^{n - 1}gm \not= 0, so f is a zerodivisor on M. Conversely, suppose f \in R is a zerodivisor on M. Consider the submodule N = \{ m \in M \mid fm = 0\} . Since N is not zero it has a weakly associated prime \mathfrak q by Lemma 10.66.5. Clearly f \in \mathfrak q and by Lemma 10.66.4 \mathfrak q is a weakly associated prime of M. \square
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