\begin{definition}[Definition:Linear Ring Action/Left]
Let $R$ be a ring.
Let $M$ be an abelian group.
A '''(left) linear ring action''' of $R$ on $M$ is a mapping from the cartesian product $\circ : R \times M \to M$ such that:
{{begin-axiom}}
{{axiom | n = 1
        | q = \forall \lambda \in R: \forall m, n \in M
        | ml= \lambda \circ \paren {m + n}
        | mo= =
        | mr= \paren {\lambda \circ m} + \paren {\lambda \circ n}
}}
{{axiom | n = 2
        | q = \forall \lambda, \mu \in R: \forall m \in M
        | ml= \paren {\lambda + \mu} \circ m
        | mo= =
        | mr= \paren {\lambda \circ m} + \paren {\mu \circ m}
}}
{{axiom | n = 3
        | q = \forall \lambda, \mu \in R: \forall m \in M
        | ml= \paren {\lambda \mu} \circ m
        | mo= =
        | mr= \lambda \circ \paren {\mu \circ m}
}}
{{end-axiom}}
\end{definition}