Basic Results about Modules

Theorem
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:

Also see

 * Basic Results about Unitary Modules