Let φ: Z "circle +" Z → Z "circle +" Z be a module homomorphism (of Z-modules). Show that if φ is surjective, it must be injective. Give an example to show that the converse is false ─ a difference between free Z-modules and vector spaces. (You may, of course, think of φ as a 2 x 2 matrix with integer entries.)
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We consider two special values , , where . Then I claim that for any , we have .
Since is a module homomorphism, then we have
Thus , for any .
Now we show that if is a surjective, then ...
This provides a proof that if a module homomorphism is surjective it is injective.