Basic_Results_about_Modules

Let $\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\struct {R, +_R, \times_R}$ be a ring whose zero is $0_R$. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $x \in G, \lambda \in R, n \in \Z$. Let $\sequence {x_m}$ be a sequence of elements of $G$. Let $\sequence {\lambda_m}$ be a sequence of elements of $R$ that is, scalars. Then