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 $$x \in G, n \in \mathbb{Z}$$.

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

Proof

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

Follows directly from Basic Results about Modules (2).


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

Follows directly from Basic Results about Modules (5).