Axiom:Left Module Axioms

Definition

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

A left module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which satisfies the following conditions:

$(\text M 1)$	$:$	Scalar Multiplication (Left) Distributes over Module Addition	$\ds \forall \lambda \in R: \forall x, y \in G:$	$\ds \lambda \circ \paren {x +_G y} $	$\ds = $	$\ds \paren {\lambda \circ x} +_G \paren {\lambda \circ y} $
$(\text M 2)$	$:$	Scalar Multiplication (Right) Distributes over Scalar Addition	$\ds \forall \lambda, \mu \in R: \forall x \in G:$	$\ds \paren {\lambda +_R \mu} \circ x $	$\ds = $	$\ds \paren {\lambda \circ x} +_G \paren {\mu \circ x} $
$(\text M 3)$	$:$	Associativity of Scalar Multiplication	$\ds \forall \lambda, \mu \in R: \forall x \in G:$	$\ds \paren {\lambda \times_R \mu} \circ x $	$\ds = $	$\ds \lambda \circ \paren {\mu \circ x} $

These stipulations are called the left module axioms.

Some sources do not distinguish between a left module and a right module, and instead refer to the left module axioms as the module axioms.

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): module
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): module

\((\text M 1)\)	$:$	Scalar Multiplication (Left) Distributes over Module Addition	\(\ds \forall \lambda \in R: \forall x, y \in G:\)	\(\ds \lambda \circ \paren {x +_G y} \)	\(\ds = \)	\(\ds \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \)
\((\text M 2)\)	$:$	Scalar Multiplication (Right) Distributes over Scalar Addition	\(\ds \forall \lambda, \mu \in R: \forall x \in G:\)	\(\ds \paren {\lambda +_R \mu} \circ x \)	\(\ds = \)	\(\ds \paren {\lambda \circ x} +_G \paren {\mu \circ x} \)
\((\text M 3)\)	$:$	Associativity of Scalar Multiplication	\(\ds \forall \lambda, \mu \in R: \forall x \in G:\)	\(\ds \paren {\lambda \times_R \mu} \circ x \)	\(\ds = \)	\(\ds \lambda \circ \paren {\mu \circ x} \)