# Definition:Linear Ring Action

## Definition

Let $R$ be a ring.

Let $M$ be an abelian group.

### Left Ring Action

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 $\circ$ satisfies the left ring action axioms:

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

### Right Ring Action

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

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

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

## Also known as

A left ring action is also known as a ring action.

## Also see

• Results about linear ring actions can be found here.