I've started self-studying algebra. So I want to err on the side of getting guidance so I don't get off on the wrong track. This is problem 2.4.4 in Artin.(adsbygoogle = window.adsbygoogle || []).push({});

Describe all homomorphisms from Z+ to Z+ (all integers under addition). Determine if they are injective, surjective, or isomorphisms.

So I need ALL the functions f s.t. f(x+y) = f(x) + f(y) for all integers x,y.

Clearly any linear function f will do this, and these are all isomorphisms.

Also f(x) = 0 for all x satisfies the definition of the homomorphism. This is not injective, surjective, nor an isomorphism.

So far so good?

I don't know of any way to prove that there are no other such functions that will satisfy the definition of a homomorphism.

Any hints?

thanks

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# All homomorphisms on Z to Z

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