Scalar Product with Multiple of Unity

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 $x \in G, n \in \Z$.

Then:

$\paren {n \cdot 1_R} \circ x = n \cdot x$

that is:

$\paren {\map {\paren {+_R}^n} {1_R} } \circ x = \map {\paren {+_G}^n} x$

Proof

Follows directly from Scalar Product with Product.

$\blacksquare$

Also see

• Basic Results about Modules
• Basic Results about Unitary Modules

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Theorem $26.2 \ (7)$