627: Generator of Module

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definition of generator of module

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Topics

About: module

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of %ring name% module.

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Target Context

• The reader will have a definition of generator of module.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$R$: $\in \{\text{ the rings }\}$
$M$: $\in \{\text{ the modules over }R\}$
$\ast S$: $\subseteq M$
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Conditions:
$\mathrm{\forall }p\in M(\mathrm{\exists }{S}^{\prime }\in \{\text{ the finite subsets of }S\},\mathrm{\exists }{r}^j\in R(p=\sum _{b_j\in S}{r}^j{b}_j))$
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$S$ has to be a finite subset of $B$, because otherwise, $p=\sum _{b_j\in S}{r}^j{b}_j$ would not make sense without $M$ equipped with any norm: definition of convergence of infinite series requires a norm.

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2: Natural Language Description

For any ring, $R$, and any module, $M$, over $R$, any (possibly uncountable) subset, $S\subseteq M$, such that each element of $M$ is a linear combination of some (finite) elements of $S$

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3: Note

Any module has a generator, because the module itself is a generator.

Each basis is a generator, but a generator is not necessarily a basis, because the generator may not be linearly independent.

Furthermore, a generator may not be able to be restricted to any basis: for example, let $R:=\mathbb{Z}$; $M:=\mathbb{Z}/(2\mathbb{Z})$, with the operations, $[z_1]+[z_2]=[z_1+z_2]$, well-defined because $[z_1+2k+z_2+2l]=[z_1+z_2+2(k+l)]=[z_1+z_2]$, and $z_1[z_2]=[z_1{z}_2]$, well-defined because $[z_1(z_2+2k)]=[z_1{z}_2+2z_{1}k]=[z_1{z}_2]$: $M=\{[0],[1]\}$. $M$ is indeed a module: 1) $[m_1]+[m_2]=[m_1+m_2]\in M$; 2) $[m_1]+[m_2]=[m_1+m_2]=[m_2+m_1]=[m_2]+[m_1]$; 3) $([m_1]+[m_2])+[m_3]=[m_1+m_2]+[m_3]=[m_1+m_2+m_3]=[m_1]+[m_2+m_3]=[m_1]+([m_2]+[m_3])$; 4) $[m]+[0]=[m+0]=[m]$; 5) $[1]+[1]=[1+1]=[2]=[0]$ and $[0]+[0]=[0+0]=[0]$; 6) $r[m]=[rm]\in M$; 7) $(r_1+r_2)[m]=[(r_1+r_2)m]=[r_{1}m+r_{2}m]=[r_{1}m]+[r_{2}m]=r_1[m]+r_2[m]$; 8) $r([m_1]+[m_2])=r[m_1+m_2]=[r(m_1+m_2)]=[r{m}_1+r{m}_2]=[r{m}_1]+[r{m}_2]=r[m_1]+r[m_2]$; 9) $(r_1{r}_2)[m]=[(r_1{r}_2)m]=[r_1(r_{2}m)]=r_1[r_{2}m]=r_1(r_2[m])$; 10) $1[m]=[1m]=[m]$. Then, $\{[1]\}$ is a generator, because $[0]=2[1]=[2]=[0]$ and $[1]=1[1]$, but the generator is not linearly independent, because $2[1]=[0]$, so, is not any basis, or cannot be restricted to be any basis.

As any vectors space is a module, 'generator of vectors space' is nothing but a generator of module.

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References

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