Category:Definitions/Examples of Modules

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This category contains definitions of examples of Module.


Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.


A module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which is both a left module and a right module:


Left Module

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.


A left module over $R$ is an $R$-algebraic structure $\struct {G, +_G, \circ}_R$ with one operation $\circ$, the (left) ring action, which satisfies the left module axioms:

\((\text M 1)\)   $:$   Scalar Multiplication (Left) Distributes over Module Addition      \(\ds \forall \lambda \in R: \forall x, y \in G:\)    \(\ds \lambda \circ \paren {x +_G y} \)   \(\ds = \)   \(\ds \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \)      
\((\text M 2)\)   $:$   Scalar Multiplication (Right) Distributes over Scalar Addition      \(\ds \forall \lambda, \mu \in R: \forall x \in G:\)    \(\ds \paren {\lambda +_R \mu} \circ x \)   \(\ds = \)   \(\ds \paren {\lambda \circ x} +_G \paren {\mu \circ x} \)      
\((\text M 3)\)   $:$   Associativity of Scalar Multiplication      \(\ds \forall \lambda, \mu \in R: \forall x \in G:\)    \(\ds \paren {\lambda \times_R \mu} \circ x \)   \(\ds = \)   \(\ds \lambda \circ \paren {\mu \circ x} \)      


Right Module

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 $\struct {G, +_G, \circ}_R$ with one operation $\circ$, the (right) ring action, which satisfies the right module axioms:

\((\text {RM} 1)\)   $:$   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      \(\ds \forall \lambda, \mu \in R: \forall x \in G:\)    \(\ds x \circ \paren {\lambda \times_R \mu} \)   \(\ds = \)   \(\ds \paren {x \circ \lambda} \circ \mu \)