Annihilator ideal

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

Annihilator ideal

See also