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Dummit and Foote open their section (part of section 10.5) on projective modules as follows:View attachment 2463D&F then deal with the issue of obtaining a homomorphism from D to M given a homomorphism from D to L and then move to the more problematic issue of obtaining a homomorphism from D to M given a homomorphism from D to N. (Strangely they refer to N as "the quotient N?). The relevant text reads as follows:
View attachment 2464
D&F then give an example ... and my question pertains to this example ... the example reads as follows:
View attachment 2465In this example, D&F make the following statement:
"Any homomorphism $$F$$ of $$D$$ into $$M = \mathbb{Z} $$must map $$D$$ to $$0$$ (since $$D$$ has no elements of order $$2$$)"
Can someone please explain why this statement is true?
I am aware that isomorphisms map elements of a given order onto elements of the same order, but here we are only dealing with a homomorphism.
Also $$0$$ does not have order $$2$$ anyway!
Can someone please clarify these issues?
Peter
View attachment 2464
D&F then give an example ... and my question pertains to this example ... the example reads as follows:
View attachment 2465In this example, D&F make the following statement:
"Any homomorphism $$F$$ of $$D$$ into $$M = \mathbb{Z} $$must map $$D$$ to $$0$$ (since $$D$$ has no elements of order $$2$$)"
Can someone please explain why this statement is true?
I am aware that isomorphisms map elements of a given order onto elements of the same order, but here we are only dealing with a homomorphism.
Also $$0$$ does not have order $$2$$ anyway!
Can someone please clarify these issues?
Peter