Definition:Linear Ring Action

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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.