Basic Results about Unitary Modules
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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:
Scalar Product with Inverse Unity
- $\paren {-1_R} \circ x = - x$
Scalar Product with Multiple of Unity
- $\paren {n \cdot 1_R} \circ x = n \cdot x$
that is:
- $\paren {\map {\paren {+_R}^n} {1_R} } \circ x = \map {\paren {+_G}^n} x$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Theorem $26.2 \ (6) - (7)$