Scalar Product with Identity

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Let $\struct {G, +_G}$ be an abelian group whose identity is $e$.

Let $\struct {R, +_R, \times_R}$ be a ring whose zero is $0_R$.

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

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


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


From Module Axiom $\text M 1$: Distributivity over Module Addition, $y \to \lambda \circ y$ is an endomorphism of $\struct {G, +_G}$.

From Module Axiom $\text M 2$: Distributivity over Scalar Addition, $\mu \to \mu \circ x$ is a homomorphism from $\struct {R, +_R}$ to $\struct {G, +_G}$.

The result follows from Homomorphism with Cancellable Codomain Preserves Identity.