Remark 10.66.10. Let \varphi : R \to S be a ring map. Let M be an S-module. Then it is not always the case that \mathop{\mathrm{Spec}}(\varphi )(\text{WeakAss}_ S(M)) \subset \text{WeakAss}_ R(M) contrary to the case of associated primes (see Lemma 10.63.11). An example is to consider the ring map
R = k[x_1, x_2, x_3, \ldots ] \to S = k[x_1, x_2, x_3, \ldots , y_1, y_2, y_3, \ldots ]/ (x_1y_1, x_2y_2, x_3y_3, \ldots )
and M = S. In this case \mathfrak q = \sum x_ iS is a minimal prime of S, hence a weakly associated prime of M = S (see Lemma 10.66.8). But on the other hand, for any nonzero element of S the annihilator in R is finitely generated, and hence does not have radical equal to R \cap \mathfrak q = (x_1, x_2, x_3, \ldots ) (details omitted).
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