Axiom:Linear Ring Action Axioms

Definition

Let $R$ be a ring.

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

$\circ$ satisfies the left linear ring action axioms if and only if:

$(1)$	$:$	$\ds \forall \lambda \in R: \forall m, n \in M:$	$\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} $

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

$\circ$ satisfies the right linear ring action axioms if and only if:

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

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

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