Endomorphism ring

Let $𝑅$ be a ring and $𝑉$ be an $𝑅$ -module. Then $E n d 𝑅 𝑉 = 𝑅 𝖬 𝗈 𝖽 (𝑉, 𝑉)$ forms a ring called the endomorphism ring, under composition, #m/def/module so for $𝑓, 𝑔 \in E n d 𝑅 𝑉$ and $𝑣 \in 𝑉$

(𝑓 + 𝑔) (𝑣) = 𝑓 (𝑣) + 𝑔 (𝑣) (𝑓 \cdot 𝑔) (𝑣) = 𝑓 \circ 𝑔 (𝑣)

If $𝑅$ is a commutative ring this becomes an $K$ -monoid, so for $𝜆, 𝜇 \in 𝑅$

(𝜆 𝑓 + 𝜇 𝑔) (𝑣) = 𝜆 𝑓 (𝑣) + 𝜇 𝑔 (𝑣)

Proof

Let $𝑓, 𝑔, ℎ \in E n d 𝑅 𝑉$ Clearly

(𝑓 + 𝑔) ℎ = 𝑓 \circ ℎ + 𝑔 \circ ℎ, ℎ (𝑓 + 𝑔) = ℎ \circ 𝑓 + ℎ \circ 𝑔 .

if $𝑅$ is commutative then

(𝛼 𝑓) \circ (𝛽 𝑔) = 𝛼 𝛽 (𝑓 \circ 𝑔)

as required.

Properties

Cayley's theorem for rings

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