933: Range of Linear Map Between Modules Is Submodule of Codomain

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description/proof of that range of linear map between modules is submodule of codomain

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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: Proof

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Starting Context

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

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Target Context

• The reader will have a description and a proof of the proposition that the range of any linear map between any modules is a submodule of the codomain.

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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:
$R$: $\in \{\text{ the rings }\}$
$M_1$: $\in \{\text{ the }R\text{ modules }\}$
$M_2$: $\in \{\text{ the }R\text{ modules }\}$
$f$: $:M_1\to M_2$, $\in \{\text{ the linear maps }\}$


Statements:
$f(M_1)\in \{\text{ the submodules of }{M}_2\}$
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2: Proof

Whole Strategy: Step 1: see that $f(M_1)$ satisfies the requirements to be a module.

Step 1:

1) $\mathrm{\forall }f(m),f(m^{\prime })\in f(M_1)(f(m)+f(m^{\prime })\in f(M_1))$ (closed-ness under addition): $f(m)+f(m^{\prime })=f(m+m^{\prime })\in f(M_1)$.

2) $\mathrm{\forall }f(m),f(m^{\prime })\in f(M_1)(f(m)+f(m^{\prime })=f(m^{\prime })+f(m))$ (commutativity of addition): it holds on ambient $M_2$.

3) $\mathrm{\forall }f(m),f(m^{\prime }),f(m^{″})\in f(M_1)((f(m)+f(m^{\prime }))+f(m^{″})=f(m)+(f(m^{\prime })+f(m^{″})))$ (associativity of additions): it holds on ambient $M_2$.

4) $\mathrm{\exists }0\in f(M_1)(\mathrm{\forall }f(m)\in f(M_1)(f(m)+0=f(m)))$ (existence of 0 element): $f(0)=0\in f(M_1)$ is the one.

5) $\mathrm{\forall }f(m)\in f(M_1)(\mathrm{\exists }f(m^{\prime })\in f(M_1)(f(m^{\prime })+f(m)=0))$ (existence of inverse element): $f(-m)=-f(m)\in f(V_1)$ is the one.

6) $\mathrm{\forall }f(m)\in f(M_1),\mathrm{\forall }r\in R(r.f(m)\in f(M_1))$ (closed-ness under scalar multiplication): $f(rm)=r.f(m)\in f(M_1)$.

7) $\mathrm{\forall }f(m)\in f(M_1),\mathrm{\forall }{r}_1,r_2\in R((r_1+r_2).f(m)=r_1.f(m)+r_2.f(m))$ (scalar multiplication distributability for scalars addition): it holds on ambient $M_2$.

8) $\mathrm{\forall }f(m),f(m^{\prime })\in f(M_1),\mathrm{\forall }r\in R(r.(f(m)+f(m^{\prime }))=r.f(m)+r.f(m^{\prime }))$ (scalar multiplication distributability for elements addition): it holds on ambient $M_2$.

9) $\mathrm{\forall }f(m)\in f(M_1),\mathrm{\forall }{r}_1,r_2\in R((r_1{r}_2).f(m)=r_1.(r_2.f(m)))$ (associativity of scalar multiplications): it holds on ambient $M_2$.

10) $\mathrm{\forall }f(m)\in f(M_1)(1.f(m)=f(m))$ (identity of 1 multiplication): it holds on ambient $M_2$.

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References

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