Dual annihilator submodule

Let $𝑉$ be a (left) $𝑅$ -module and $𝑉 *$ be its Dual module. Given a subset $𝑆 \subseteq 𝑉$ , the annihilator of $𝑆$ in $𝑉 *$ is the set of all linear functionals which annihilate $𝑆$ , #m/def/module i.e.

𝑉 * A n n (𝑆) : = {𝑓 \in 𝑉 * : ∣ : ⟨ 𝑓, 𝑆 ⟩ = 0} .

Clearly this is an $𝑅$ -submodule of $𝑉 *$ . For the remainder of this Zettel (and elsewhere) the annihilator of $𝑆$ is denoted $𝑆 ⟂$ , and this concept does in a sense generalize the orthogonal complement.

Properties

The following hold when $𝑅 = 𝕂$ is a field:

For $𝑊 \leq 𝕂 𝑉$ , we have $⟨ 𝑊 ⟂, 𝑥 ⟩ = 0$ iff $𝑥 \in 𝑊$ .
For $𝑋 \leq 𝕂 𝑉$ and $𝑌 \leq 𝕂 𝑊$ , we have $(𝑋 \otimes 𝑌) ⟂ ≅ 𝑋 ⟂ \otimes 𝑊 * + 𝑉 \otimes 𝑌 ⟂$ .

Proof of 1

By definition, if $𝑥 \in 𝑊$ then $⟨ 𝑊 ⟂, 𝑥 ⟩ = 0$ . For the converse, suppose towards contradiction that $⟨ 𝑊 ⟂, 𝑥 ⟩ = 0$ but $𝑥 \notin 𝑊$ . Let $𝑍$ be a complement subspace to $𝑊 \oplus 𝕂 𝑧$ , and define $𝑓$ so that $𝑓 (𝑍 \oplus 𝑊) = 0$ and $𝑓 (𝑥) = 1$ . Then $𝑓 \in 𝑊 ⟂$ , a contradiction. Thus ^F1 is proven. .

Proof of 2

Taking duals of the inclusions $𝜄 𝑋 : 𝑋 ↪ 𝑉$ and $𝜄 𝑌 : 𝑌 ↪ 𝑊$ gives maps $𝜄 * 𝑋 𝑉 * ↠ 𝑋 *$ and $𝜄 * 𝑌 : 𝑊 * ↠ 𝑌 *$ , where $𝑋 ⟂ = k e r 𝜄 * 𝑋$ and $𝑌 ⟂ = 𝜄 * 𝑌$ . The universal property of kernels gives factorizations

for some monomorphisms $𝑗 1, 𝑗 2$ .

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