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I would like to come up with an example for a module homomorphism between two finitely generated, not free modules over $\mathbb{Z}[i]$. Also, what would be the kernel and image in this example?

user223365
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  • What do you know about homomorphisms? In other words, what property would such a function have to satisfy? – Sammy Black Mar 13 '15 at 18:10
  • I know that addition and multiplication by a scalar would be preserved for a homomorphism, but I'm having trouble coming up with a specific example that fits all those conditions. – user223365 Mar 13 '15 at 18:17

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How about the identity map $M\to M$, where $M$ is any non-free, finitely generated $\mathbb{Z}[i]$-module?

Do you know how to construct such a thing? Hint: It may be easier to first consider $\mathbb{Z}$.

Andrew Dudzik
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  • Are there any examples with two distinct modules that would work? – user223365 Mar 13 '15 at 18:46
  • @user223365 Yes. But it would be good to start with one... always work on the easiest problem that you don't know how to do. – Andrew Dudzik Mar 13 '15 at 18:46
  • I see, are there any suggestions you can offer for how to produce such a module M? – user223365 Mar 13 '15 at 18:53
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    Would a cyclic module M = Ra work? Would it be non-free? – user223365 Mar 13 '15 at 19:01
  • @user223365 It might be best to work with a ring you are familiar with, such as $\mathbb{Z}$. What are the cyclic $\mathbb{Z}$-modules? If you can find two, $M$ and $N$, then the zero map $M\to N$ is a homomorphism between two non-free modules. – Andrew Dudzik Mar 14 '15 at 02:53