# Category:Linear Ring Actions

This category contains results about Linear Ring Actions.
Definitions specific to this category can be found in Definitions/Linear Ring Actions.

Let $R$ be a ring.

Let $M$ be an abelian group.

### Left Ring Action

A (left) linear ring action of $R$ on $M$ is a mapping from the cartesian product $\circ : R \times M \to M$ such that:

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

### Right Ring Action

A right linear ring action of $R$ on $M$ is a mapping from the cartesian product $\circ : M \times R \to M$ such that:

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

This category currently contains no pages or media.