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I'm studying projective and injective modules and a serious question raised for me: what is the necessary and sufficient conditions for existence of homomorphisms between modules?

  • I mean non-trivial case.
  • I know it is very broad question. But at least it might have been categorized to some extent.
  • Classification is one of the most important things in mathematics so it could be an interesting material for research.
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    $\begingroup$ Well there's always the zero homomorphism ... Little more can be said in general $\endgroup$
    – Hagen von Eitzen
    Sep 6, 2015 at 17:52
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    $\begingroup$ This is far, far too broad. $\endgroup$
    – Matt Samuel
    Sep 6, 2015 at 17:57
  • $\begingroup$ Why too broad? It's a reasonable question to ask and we know how to answer it for vector spaces over a field. So what can we say about modules over other rings? $\endgroup$
    – Rob Arthan
    Sep 6, 2015 at 23:22

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