Definition:Left Module Axioms

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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:

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

These stipulations are called the left module axioms.

Also known as

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

Also see