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