Simple, Cyclic and Projective Modules I have been taking a course in Homological Algebra and revisiting the lecture notes for some reason. There was a non-answered question about showing two non zero modules A,B over a non trivial unital ring(in fact upper triangular matrices) are projective modules. I won't explain what these modules are because it requires a lot of work.
I thought that if there is a relation between being cyclic and/or simple AND projective then I can use that. However, I couldn't find any such relation.
Is there any relation between cyclic and/or simple modules and projective modules?


I thought that if there is a relation between being cyclic and/or simple AND projective then I can use that.


*

*A simple module is always cyclic ( every nonzero element generates it.)


*A projective cyclic module need not be simple (for example $R_R$ for any ring that isn’t a division ring.)


*A simple module need not be projective.  (For example, $\mathbb Z/2\mathbb Z$ is not projective as a $\mathbb Z$ module.)
I think this takes care of all the possible mixtures of relationships that you could test.  There is very little to say positively.
