Basic Results about Modules
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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
- $\lambda \circ \paren {n \cdot x} = n \cdot \paren {\lambda \circ x} = \paren {n \cdot \lambda} \circ x$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Theorem $26.2 \ (1) - (5)$