Scalar Product with Inverse Unity
Jump to navigation
Jump to search
Theorem
Let $\struct {G, +_G}$ be an abelian group whose identity is $e$.
Let $\struct {R, +_R, \times_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let $\struct {G, +_G, \circ}_R$ be an unitary $R$-module.
Let $x \in G$.
Then:
- $\paren {-1_R} \circ x = - x$
Proof
Follows directly from Scalar Product with Inverse.
$\blacksquare$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Theorem $26.2 \ (6)$