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I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules.
I am studying Proposition 28 (D&F pages 387 - 388)
In the latter stages of the proof of Proposition 28 we find the following statement (top of page 388):
"In general, $$ Hom_R (R, X) \cong X $$, the isomorphism being given by mapping a homomorphism to its value on the element $$1 \in R $$"
I am having some trouble in demonstrating the isomorphism involved in the relationship $$ Hom_R (R, X) \cong X $$.
To demonstrate the isomorphism I proceeded as follows:
Let $$f, g \in Hom_R (R,X) $$ so $$ f,g : \ R \to X $$
Consider $$ \theta \ : \ Hom_R (R,X) \to X $$
where $$ \theta (f) = f(1_R) $$
To show $$ \theta $$ is a homomorphism we proceed as follows:
$$ \theta (f + g) = (f + g)(1_R) = f(1_R) + g(1_R) $$
$$ = \theta (f) + \theta (g) $$
and
$$ \theta (rf) = (rf) = rf (1_R) = r \theta (f) $$ where $$ r \in R $$
Then I need to show $$ \theta $$ is injective and surjective.
BUT ... I am having problems in demonstrating that $$ \theta $$ is injective ... can someone help me with this task?Note that I suspect one would proceed as follows:
Suppose we have $$ f, g \in Hom_R (R,X) $$ such that:
$$ \theta (f) = f(1_R) $$ and $$ \theta (g) = f(1_R) $$
Now we have, of course, by definition of g, that $$ \theta (g) = g(1_R) $$
So $$ f(1_R) = g(1_R) $$ ... but how do we proceed from here to show that f = g?
Hope someone can help.
Peter
I am studying Proposition 28 (D&F pages 387 - 388)
In the latter stages of the proof of Proposition 28 we find the following statement (top of page 388):
"In general, $$ Hom_R (R, X) \cong X $$, the isomorphism being given by mapping a homomorphism to its value on the element $$1 \in R $$"
I am having some trouble in demonstrating the isomorphism involved in the relationship $$ Hom_R (R, X) \cong X $$.
To demonstrate the isomorphism I proceeded as follows:
Let $$f, g \in Hom_R (R,X) $$ so $$ f,g : \ R \to X $$
Consider $$ \theta \ : \ Hom_R (R,X) \to X $$
where $$ \theta (f) = f(1_R) $$
To show $$ \theta $$ is a homomorphism we proceed as follows:
$$ \theta (f + g) = (f + g)(1_R) = f(1_R) + g(1_R) $$
$$ = \theta (f) + \theta (g) $$
and
$$ \theta (rf) = (rf) = rf (1_R) = r \theta (f) $$ where $$ r \in R $$
Then I need to show $$ \theta $$ is injective and surjective.
BUT ... I am having problems in demonstrating that $$ \theta $$ is injective ... can someone help me with this task?Note that I suspect one would proceed as follows:
Suppose we have $$ f, g \in Hom_R (R,X) $$ such that:
$$ \theta (f) = f(1_R) $$ and $$ \theta (g) = f(1_R) $$
Now we have, of course, by definition of g, that $$ \theta (g) = g(1_R) $$
So $$ f(1_R) = g(1_R) $$ ... but how do we proceed from here to show that f = g?
Hope someone can help.
Peter
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