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I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules.
I am studying Proposition 29 (D&F, page 388)
I need some help in order to fully understand the proof of the last statement of Proposition 29.
Proposition 29 and its proof (Ch 10, D&F page 388) reads as follows:
View attachment 2494
As can be seen above, in the proof, D&F define $$ \pi_1 \pi_2, f, \pi_1 \circ f text{ and } \pi_2 \circ f $$ and then state the following:
"This defines a map from $$ Hom_R (D, L \oplus N) $$ to $$ Hom_R ( D, L ) \oplus Hom_R ( D, N )$$ which is easily seen to be a homomorphism.
Can someone please help me to see explicitly and formally how the definitions of MATH] \pi_1 \pi_2, f, \pi_1 \circ f text{ and } \pi_2 \circ f $$ lead to such a map and also help me to determine an explicit expression/formala for this homomorphism?
Hope someone can help.
Peter
I am studying Proposition 29 (D&F, page 388)
I need some help in order to fully understand the proof of the last statement of Proposition 29.
Proposition 29 and its proof (Ch 10, D&F page 388) reads as follows:
View attachment 2494
As can be seen above, in the proof, D&F define $$ \pi_1 \pi_2, f, \pi_1 \circ f text{ and } \pi_2 \circ f $$ and then state the following:
"This defines a map from $$ Hom_R (D, L \oplus N) $$ to $$ Hom_R ( D, L ) \oplus Hom_R ( D, N )$$ which is easily seen to be a homomorphism.
Can someone please help me to see explicitly and formally how the definitions of MATH] \pi_1 \pi_2, f, \pi_1 \circ f text{ and } \pi_2 \circ f $$ lead to such a map and also help me to determine an explicit expression/formala for this homomorphism?
Hope someone can help.
Peter