## 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:

### Scalar Product with Identity

$\lambda \circ e = 0_R \circ x = e$

### Scalar Product with Inverse

$\lambda \circ \struct {-x} = \struct {-\lambda} \circ x = -\struct {\lambda \circ x}$

### Scalar Product with Sum

$\ds \lambda \circ \paren {\sum_{k \mathop = 1}^m x_k} = \sum_{k \mathop = 1}^m \paren {\lambda \circ x_k}$

### Product with Sum of Scalar

$\ds \paren {\sum_{k \mathop = 1}^m \lambda_k} \circ x = \sum_{k \mathop = 1}^m \paren {\lambda_k \circ x}$

### Scalar Product with Product

$\lambda \circ \paren {n \cdot x} = n \cdot \paren {\lambda \circ x} = \paren {n \cdot \lambda} \circ x$