# Scalar Product with Inverse

## Theorem

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {R, +_R, \times_R}$ be a ring.

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

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

Then:

$\lambda \circ \struct {-x} = \struct {-\lambda} \circ x = -\struct {\lambda \circ x}$

## Proof

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 Identity Preserves Inverses.

$\blacksquare$