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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)
I need some help in order to fully understand the proof of the last statement of Proposition 28.
Proposition 28 (Ch 10, D&F pages 387-388) reads as follows:
View attachment 2487The proof of the last statement of the proposition reads as follows:View attachment 2488
View attachment 2489
In the proof, after the statement:
"In general, $$ Hom_R (R, X) \cong X $$, the isomorphism being given by mapping a homomorphism to its value on the element $$1 \in R $$"
D&F write:
Taking D = R in (10), the exactness of the sequence:
$$ 0 \longrightarrow L \stackrel{\psi}{\longrightarrow} M \stackrel{\psi}{\longrightarrow} N $$
follows easily.
Can someone please explain how/why this follows easily? ... ... ... that is how/why does taking D=R in (10) lead easily to the result that the sequence $$ 0 \longrightarrow L \stackrel{\psi}{\longrightarrow} M \stackrel{\psi}{\longrightarrow} N $$ is exact?Further, it seems strange to me that in an exact sequence of modules we are putting D = R where (I assume) R is a ring. (Yes, I understand that a ring can be viewed as a module over itself, but it seems it is a special case, which we are using to establish a general result?) Can someone please clarify this issue?Hope someone can help.
Peter
I am studying Proposition 28 (D&F pages 387 - 388)
I need some help in order to fully understand the proof of the last statement of Proposition 28.
Proposition 28 (Ch 10, D&F pages 387-388) reads as follows:
View attachment 2487The proof of the last statement of the proposition reads as follows:View attachment 2488
View attachment 2489
In the proof, after the statement:
"In general, $$ Hom_R (R, X) \cong X $$, the isomorphism being given by mapping a homomorphism to its value on the element $$1 \in R $$"
D&F write:
Taking D = R in (10), the exactness of the sequence:
$$ 0 \longrightarrow L \stackrel{\psi}{\longrightarrow} M \stackrel{\psi}{\longrightarrow} N $$
follows easily.
Can someone please explain how/why this follows easily? ... ... ... that is how/why does taking D=R in (10) lead easily to the result that the sequence $$ 0 \longrightarrow L \stackrel{\psi}{\longrightarrow} M \stackrel{\psi}{\longrightarrow} N $$ is exact?Further, it seems strange to me that in an exact sequence of modules we are putting D = R where (I assume) R is a ring. (Yes, I understand that a ring can be viewed as a module over itself, but it seems it is a special case, which we are using to establish a general result?) Can someone please clarify this issue?Hope someone can help.
Peter
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