1018: Bijective Linear Map Between Modules Is 'Modules - Linear Morphisms' Isomorphism

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description/proof of that bijective linear map between modules is 'modules - linear morphisms' isomorphism

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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: Note
• 3: Proof

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Starting Context

• The reader knows a definition of bijection.
• The reader knows a definition of linear map.
• The reader knows a definition of %ring name% module.
The reader knows a definition of %category name% isomorphism.

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Target Context

• The reader will have a description and a proof of the proposition that any bijective linear map between any modules is a 'modules - linear morphisms' isomorphism.

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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:
$R$: $\in \{\text{ the rings }\}$
$M_1$: $\in \{\text{ the }R\text{ modules }\}$
$M_2$: $\in \{\text{ the }R\text{ modules }\}$
$f$: $:M_1\to M_2$, $\in \{\text{ the linear maps }\}\cap \{\text{ the bijections }\}$
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Statements:
$f\in \{\text{ the 'modules - linear morphisms' isomorphisms }\}$
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2: Note

In general, a bijective morphism is not necessarily an isomorphism: for example, a bijective continuous map is not necessarily a 'topological spaces - continuous maps' isomorphism, because the inverse is not necessarily continuous, which is the reason why we need to specifically prove this proposition.

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3: Proof

Whole Strategy: Step 1: define the inverse, $f^{-1}:M_2\to M_1$; Step 2: see that $f^{-1}$ is linear; Step 3: conclude the proposition.

Step 1:

As $f$ is bijective, the inverse, $f^{-1}:M_2\to M_1$, is well-defined.

Step 2:

Let us see that $f^{-1}$ is linear.

Let $m_1,m_2\in M_2$ and $r_1,r_2\in R$ be any. $f^{-1}(r_1{m}_1+r_2{m}_2)=r_1{f}^{-1}(m_1)+r_2{f}^{-1}(m_2)$? $f(r_1{f}^{-1}(m_1)+r_2{f}^{-1}(m_2))=r_{1}f(f^{-1}(m_1))+r_{2}f(f^{-1}(m_2))=r_1{m}_1+r_2{m}_2$, which means that $f^{-1}(r_1{m}_1+r_2{m}_2)=r_1{f}^{-1}(m_1)+r_2{f}^{-1}(m_2)$. So, $f^{-1}$ is linear.

Step 3:

So, $f$ is a 'modules - linear morphisms' isomorphism.

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References

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