Axiom:Right Module Axioms

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.