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If $M$ and $N$ are $R$- modules, then under what conditions $\operatorname{Hom}(M,N)$ is projective?

I was trying to show that $\operatorname{Hom}(M,N)$ might be written as tensor product of two modules, i.e. Of dual of $M$ and $N$ (like in case of vector spaces), if $M$ and $N$ are free. And then use the fact that tensor product of two free modules is free, but I was unable to extend the proof in case of vector spaces to modules. I don't know is it right direction. So any help regarding this would be appreciated...

M47145
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satyendra
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  • Hi: Welcome to math.SE. Questions that treat the site as a "homework mill for question statements" are not well received. Generally, adding any substantive work and thoughts (even if they are not successful attempts) on a question will make it an admissible question. So, what have you tried up to now? – rschwieb Aug 28 '15 at 13:01
  • I had tried considering M and N projective modules, but I don't know I am thinking right or not as I am not able to conclude anything regarding Hom (M,N). So, I just want to know are there some conditions on M and N. I have been trying it for last two days, but I was unable to do and this is not my homework. – satyendra Aug 28 '15 at 13:11
  • so I don't want proofs, I just need the possible conditions. I have not found any question regarding this, when I searched Dummit and foote. Even on google there is not any problem mentioning this I had found. So, If someone can tell this. Please help me. – satyendra Aug 28 '15 at 13:15
  • Add your thoughts to the post, not the comments. A lot of people are going to take one look at the body of your post, potentially downvote, and leave. – rschwieb Aug 28 '15 at 13:49
  • I am new. So i dont know that i have to post the arguements also. I was trying to show that Hom(M,N) might be written as tensor product of two modules, i.e. Of dual of M and N(like in case of vector spaces), if M and N are free. And then use the fact that tensor product of two free modules is free, but I was unable to extend the proof in case of vector spaces to modules. I dont know is it right direction... – satyendra Aug 28 '15 at 19:28
  • Here's something that is true, which you might like to prove: $\mathrm{Hom} (M, N)$ is projective if M is finitely generated projective and N is projective. – Zhen Lin Aug 28 '15 at 20:00
  • Thanx for the help. Can we write hom(m,n) as a tensor product of modules like vector spaces – satyendra Aug 29 '15 at 13:04

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