Editing Annihilator (ring theory) (section)

==== Short exact sequences ====
Given a [[short exact sequence]] of modules,
:<math>0 \to M' \to M \to M'' \to 0,</math>
the support property
:<math>\operatorname{Supp}M = \operatorname{Supp}M' \cup \operatorname{Supp}M'',</math><ref>{{Cite web|title=Lemma 10.39.9 (00L3)—The Stacks project|url=https://stacks.math.columbia.edu/tag/00L3|website=stacks.math.columbia.edu|access-date=2020-05-13}}</ref>
together with the relation with the annihilator implies
:<math>V(\operatorname{Ann}_R(M)) = V(\operatorname{Ann}_R(M')) \cup V(\operatorname{Ann}_R(M'')).</math>
More specifically, the relations
:<math>\operatorname{Ann}_R(M') \cap \operatorname{Ann}_R(M'') \supseteq \operatorname{Ann}_R(M) \supseteq \operatorname{Ann}_R(M') \operatorname{Ann}_R(M''). </math>
If the sequence splits then the inequality on the left is always an equality. This holds for arbitrary [[direct sum of modules|direct sums]] of modules, as
:<math>\operatorname{Ann}_R\left( \bigoplus_{i\in I} M_i \right) = \bigcap_{i\in I} \operatorname{Ann}_R(M_i).</math>