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I have lots of confusion about definition of $Hom(A,B)$. I would like to ask several questions with my thoughts. Hopefully I could solve my problem.

-Firstly, my book write that if $A$ and $B$ is R-left module then $Hom(A,B)$ is a set of module homomorphisms from $A$ to $B$. So, does it mean that $Hom(A,B)$ is a set of morphisms from $A$ to $B$. Definition of morphism is:

-$f$ is a morphism from $A$ to $B$ if $f$ preserves identity and composition (associative).

-Therefore, if $A$ is a ring or a group, $Hom (A,B)$ is a set of morphism from $A$ to $B$ then is it a ring homomorphism and group homomorphism? And when $A$ is a ring then $f$ being group homomorphism from $A$ to $B$ is in $Hom(A,B)$ because $A$ is a ring so it is also a group? Probably, I describe my questions very clumsily so hope that you could understand and help me answer it.

Jean Marie
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Thế Long Lê
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  • These are not rings, they are modules. Look up the dexinition of homomorphism of modules. – Tobias Kildetoft Apr 20 '16 at 15:55
  • I am sorry. I misunderstand what you say. Is that $A$ and $B$ could not be rings when we define $Hom(A,B)$? – Thế Long Lê Apr 20 '16 at 16:06
  • When you define $Hom(A,B)$ you have to know in advance what objects $A$ and $B$ are so yes, $A$ and $B$ might not be rings. – noctusraid Apr 20 '16 at 16:55
  • Ok, I need you confirm that again. The element in $Hom(A,B)$ when $A,B$ are rings is ring morphism, i.e ring homomorphism? What about this example: $Hom(\mathbb{Z_{n}}, \mathbb{Q})$ – Thế Long Lê Apr 21 '16 at 01:38
  • By $\mathbb Z_n$ we denote the cyclic group, so I'd guess (you see, this depends alot on the context but should be clear most of the times by the context) it's the set of group homomorphisms. As the rationals are a field one can view them as a group with respect to addition or multiplication, depending on this choice you get different group homomorphisms. – noctusraid Apr 21 '16 at 07:32
  • Why are you ignoring what you wrote yourself? The book writes that these are module homomorphisms, so this is the term you ought to look up. – Tobias Kildetoft Apr 21 '16 at 07:35
  • @TobiasKildetoft I think OP stumbled upon the definition of $hom( \cdot, \cdot)$ while studying modules in his book and is now confused by the very definition of $hom( \cdot, \cdot)$ and not about the particular case of modules. That's how I see it. – noctusraid Apr 21 '16 at 11:15
  • Yes, the fact that my book take A and B in Category(Mod), so these are R-homomorphism or R maps or module homomorphism, right? But actually, I would like to ask what happend if A,B are not in Category(Mod)? Thank you for your help :) – Thế Long Lê Apr 21 '16 at 15:42
  • @ThếLongLê right. So is your questioned settled by my answer below? – noctusraid Apr 21 '16 at 17:22
  • Yes, I got it.. – Thế Long Lê Apr 22 '16 at 03:47

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