Basic Results about Unitary Modules

Theorem
Let $\left({G, +_G}\right)$ be an abelian group whose identity is $e$.

Let $\left({R, +_R, \times_R}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

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

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

Then:
 * $(1) \quad \left({- 1_R}\right) \circ x = - x$


 * $(2) \quad \left({n \cdot 1_R}\right) \circ x = n \cdot x$, that is: $\left({\left({+_R}\right)^n \left({1_R}\right)}\right) \circ x = \left({+_G}\right)^n \left({x}\right)$

Proof

 * $(1) \quad \left({- 1_R}\right) \circ x = - x$:

Follows directly from Basic Results about Modules $(2)$.


 * $(2) \quad \left({n \cdot 1_R}\right) \circ x = n \cdot x$:

Follows directly from Basic Results about Modules $(5)$.