Annihilator

Let $𝑀$ be a (left) $𝑅$ -module and $𝑆 \subseteq 𝑀$ . The annihilator $𝑅 A n n 𝑆$ is the (left) ideal made up of all elements of $𝑅$ which annihilate $𝑆$ , module i.e.

𝑅 A n n 𝑆 = {𝑟 \in 𝑅 : 𝑟 ⊙ 𝑆 = 0} .

If $𝑆 \leq 𝑅 𝑀$ is a submodule, then $𝑅 A n n 𝑆$ is a two-sided ideal.

Proof of ideal

If $𝑟 \in 𝑅 A n n 𝑆$ then for any $𝑡 \in 𝑅$ we have
$𝑡 𝑟 ⊙ 𝑆 = 𝑡 ⊙ (𝑟 ⊙ 𝑆) = 𝑡 (0) = 0$
so $𝑡 𝑟 \in 𝑅 A n n 𝑆$ . With the additional assumption that $𝑆 \leq 𝑅 𝑀$ is a submodule, we have
$𝑟 𝑡 ⊙ 𝑆 = 𝑟 ⊙ (𝑡 ⊙ 𝑆) \subseteq 𝑟 ⊙ 𝑆 = 0$
so $𝑟 𝑡 \in 𝑅 A n n 𝑆$ as required.

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Annihilator

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