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Module homomorphism proof

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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Solution Summary

This provides a proof that if a module homomorphism is surjective it is injective.