Axiom:Linear Ring Action Axioms/Left

Definition

Let $R$ be a ring.

Let $M$ be an abelian group.

Let $\circ : R \times M \to M$ be a mapping from the cartesian product $R \times M$.

$\circ$ is a left linear ring action of $R$ on $M$ if and only if $\circ$ satisifes the axioms:

\((1)\)	$:$			\(\ds \forall \lambda \in R: \forall m, n \in M:\)		\(\ds \lambda \circ \paren {m + n} \)	\(\ds = \)	\(\ds \paren {\lambda \circ m} + \paren {\lambda \circ n} \)
\((2)\)	$:$			\(\ds \forall \lambda, \mu \in R: \forall m \in M:\)		\(\ds \paren {\lambda + \mu} \circ m \)	\(\ds = \)	\(\ds \paren {\lambda \circ m} + \paren {\mu \circ m} \)
\((3)\)	$:$			\(\ds \forall \lambda, \mu \in R: \forall m \in M:\)		\(\ds \paren {\lambda \mu} \circ m \)	\(\ds = \)	\(\ds \lambda \circ \paren {\mu \circ m} \)

These criteria are called the left linear ring action axioms.

Also see

• Axiom:Right Linear Ring Action Axioms

• Definition:Left Linear Ring Action

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