Category:Definitions/Examples of Modules
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 either a left module or a right module, the type is unspecified:
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 | \(\ds \forall \lambda, \mu \in R: \forall x \in G:\) | \(\ds x \circ \paren {\lambda \times_R \mu} \) | \(\ds = \) | \(\ds \paren {x \circ \lambda} \circ \mu \) |
Subcategories
This category has the following 3 subcategories, out of 3 total.
A
- Definitions/Artinian Modules (1 P)
N
Pages in category "Definitions/Examples of Modules"
The following 10 pages are in this category, out of 10 total.