# Scalar Product with Inverse

## Theorem

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

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

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

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

Then:

$\lambda \circ \left({- x}\right) = \left({- \lambda}\right) \circ x = - \left({\lambda \circ x}\right)$

## 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 Identity Preserves Inverses.

$\blacksquare$