592: Direct Sum of Modules

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definition of direct sum of modules

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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.
• The reader knows a definition of product set.

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Target Context

• The reader will have a definition of direct sum of modules.

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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:
$R$: $\in \{\text{ the rings }\}$
$\{M_{\alpha }\}$: $M_{\alpha }\in \{\text{ the }R\text{ modules }\}$, where $\alpha \in A$ where $A$ is any possibly uncountable index set
$\ast {\oplus }_{\alpha \in A}{M}_{\alpha }$: $=\{{\times }_{\alpha \in A}{p}_{\alpha }\in {\times }_{\alpha \in A}{M}_{\alpha }|p_{\alpha }=0\text{ for all except some finite indices}\}$ with the $R$ module operations, where ${\times }_{\alpha \in A}{M}_{\alpha }$ denotes the product set
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Conditions:
$\mathrm{\forall }r,r^{\prime }\in R,\mathrm{\forall }{\times }_{\alpha \in A}{p}_{\alpha },{\times }_{\alpha \in A}{p}_{\alpha }^{\prime }\in {\oplus }_{\alpha \in A}{M}_{\alpha }(r{\times }_{\alpha \in A}{p}_{\alpha }+r^{\prime }{\times }_{\alpha \in A}{p}_{\alpha }^{\prime }={\times }_{\alpha \in A}(r{p}_{\alpha }+r^{\prime }{p}_{\alpha }^{\prime }))$
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2: Natural Language Description

For any ring, $R$, and any $R$ modules, $\{M_{\alpha }|\alpha \in A\}$, where $A$ is any possibly uncountable index set, ${\oplus }_{\alpha \in A}{M}_{\alpha }:=\{{\times }_{\alpha \in A}{p}_{\alpha }\in {\times }_{\alpha \in A}{M}_{\alpha }|p_{\alpha }=0\text{ for all except some finite indices}\}$ with the $R$ module operations, $\mathrm{\forall }r,r^{\prime }\in R,\mathrm{\forall }{\times }_{\alpha \in A}{p}_{\alpha },{\times }_{\alpha \in A}{p}_{\alpha }^{\prime }\in {\oplus }_{\alpha \in A}{M}_{\alpha }(r{\times }_{\alpha \in A}{p}_{\alpha }+r^{\prime }{\times }_{\alpha \in A}{p}_{\alpha }^{\prime }={\times }_{\alpha \in A}(r{p}_{\alpha }+r^{\prime }{p}_{\alpha }^{\prime }))$, where ${\times }_{\alpha \in A}{M}_{\alpha }$ denotes the product set

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3: Note

When $A$ is finite, this definition equals 'direct product of structures' for which the structures are modules. This definition cannot be generalized to general structures because $0$ is not generally defined.

Do not confuse this definition with 'group as direct sum of finite number of normal subgroups', which is not about creating a new group from the constituent groups. As any Abelian group is a module, there can be the direct sum of some Abelian groups by the concept of this definition, but that direct sum of modules is not exactly the direct sum of the constituent groups by 'group as direct sum of finite number of normal subgroups': any element of $G_1\oplus G_2$ is of the form, $(p_1,p_2)$, while any element of $G_1$, $p_1$, is not of the form, so, is not any element of $G_1\oplus G_2$, so, $G_1$ is not any subgroup of $G_1\oplus G_2$; in fact, $G$ is the direct sum of $G_1\times \{1\}$ and $\{1\}\times G_2$ by 'group as direct sum of finite number of normal subgroups'.

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References

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