Definition:Linear Ring Action/Left
< Definition:Linear Ring Action(Redirected from Definition:Left Linear Ring Action)
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Definition
Let $R$ be a ring.
Let $M$ be an abelian group.
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:
\((1)\) | $:$ | \(\ds \forall \lambda \in R: \forall m, n \in M:\) | \(\ds \lambda \circ \paren {m + n} \) | \(\ds = \) | \(\ds \paren {\lambda \circ m} + \paren {\lambda \circ n} \) | ||||
\((2)\) | $:$ | \(\ds \forall \lambda, \mu \in R: \forall m \in M:\) | \(\ds \paren {\lambda + \mu} \circ m \) | \(\ds = \) | \(\ds \paren {\lambda \circ m} + \paren {\mu \circ m} \) | ||||
\((3)\) | $:$ | \(\ds \forall \lambda, \mu \in R: \forall m \in M:\) | \(\ds \paren {\lambda \mu} \circ m \) | \(\ds = \) | \(\ds \lambda \circ \paren {\mu \circ m} \) |
Also known as
A left ring action is also known as a ring action.
Also see
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