# If $M$and $N$ are R- modules, then under what conditions $\operatorname{Hom}(M,N)$ the space of R-module morphisms from M to N, is projective?

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...

• 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