Basic Results about 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 whose zero is $0_R$.

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

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

Let $\left \langle {x_m} \right \rangle$ be a sequence of elements of $G$.

Let $\left \langle {\lambda_m} \right \rangle$ be a sequence of elements of $R$ i.e. scalars.

Then: