Scalar Product with Identity

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Theorem

Let $\left({G, +_G}\right)$ be an abelian group whose identity is $e$.

Let $\left({R, +_R, \times_R}\right)$ be a ring whose zero is $0_R$.

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module.


Let $x \in G, \lambda \in R$.


Then:

$\lambda \circ e = 0_R \circ x = e$


Proof

From Module: $(1)$, $y \to \lambda \circ y$ is an endomorphism of $\left({G, +_G}\right)$.

From Module: $(2)$, $\mu \to \mu \circ x$ is a homomorphism from $\left({R, +_R}\right)$ to $\left({G, +_G}\right)$.

The result follows from Homomorphism with Cancellable Codomain Preserves Identity.

$\blacksquare$


Sources