Axiom:Right Module Axioms
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Definition
Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {G, +_G}$ be an abelian group.
A right module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which satisfies the following conditions:
\((\text {RM} 1)\) | $:$ | Scalar Multiplication Right Distributes over Module Addition | \(\ds \forall \lambda \in R: \forall x, y \in G:\) | \(\ds \paren {x +_G y} \circ \lambda \) | \(\ds = \) | \(\ds \paren {x \circ \lambda} +_G \paren {y \circ \lambda} \) | |||
\((\text {RM} 2)\) | $:$ | Scalar Multiplication Left Distributes over Scalar Addition | \(\ds \forall \lambda, \mu \in R: \forall x \in G:\) | \(\ds x \circ \paren {\lambda +_R \mu} \) | \(\ds = \) | \(\ds \paren {x \circ \lambda} +_G \paren {x\circ \mu} \) | |||
\((\text {RM} 3)\) | $:$ | Associativity of Scalar Multiplication | \(\ds \forall \lambda, \mu \in R: \forall x \in G:\) | \(\ds x \circ \paren {\lambda \times_R \mu} \) | \(\ds = \) | \(\ds \paren {x \circ \lambda} \circ \mu \) |
These stipulations are called the right module axioms.