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