I have been studying modules and homological algebra as of late but somehow I have missed how to calculate Hom(A,B) for abelian groups, modules and Hom(A,_)/Hom(_,B) for exact sequences. I have no problem with the abstract nonsense in the subject. But explicit examples are useful and this construction is seldom mentioned explicitly in texts so I suspect it's pretty trivial but I'd still like someone to write this out for me.
Is it as as simple as finding generators in A and picking elements in B to send them to? (with careful adjectives of course, but I am looking for the general idea.)
Also some examples would be really helpful! (If they show a general idea, not just some clever construction.)
Thanks!