Definition:Linear Ring Action

This page is about Linear Ring Action. For other uses, see 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

• Definition:Module over Ring
• Correspondence between Linear Ring Actions and Ring Representations

• Results about linear ring actions can be found here.

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