Should a researcher share scientific ideas in a temporary position and are they entitled to recognition for doing so? A postdoc joins a group with a good idea to work on, while no specific project was given to him by the PI. After recognising the potential of the idea, the PI decides to take PhD students to work on it. When the postdoc asks, the PI says that the postdoc is going to be on this position only for a short time and the project (work) may not be finished before his contract expires. The contract is funded by the PI.

What should the postdoc do? Is it right for the postdoc to give his ideas in the first instance and should he be recognised when the work gets completed? If so, what about authorship?
When I was a graduate student, my advisor told me something that has stuck with me ever since:


  I never worry about giving my ideas away.  I always have more ideas than I can work on, so I give them away as fast as I can.  The only people who worry about somebody stealing their ideas away are people who are afraid they won't have any more.


I have taken that to heart in my career: I give (most) ideas away as fast as I can, and it has served me well.  Having ideas is not an art or a mystical experience, it is the exercise of a skill at creative thinking that can be developed and strengthened just like any other skill.  Over time, giving ideas away thus doesn't diminish your stock of ideas, but instead actually increases it.

The real question, as alluded to by others, is what you do with an idea once you have it.  As I see it, there are basically three honorable actions that you can take (hoarding it away not being one of them):


Pursue the work following from the idea strongly yourself, heavily investing your own time.  In this case, if the work is fruitful, you most certainly should get a lot of credit.  These are the only ideas I might not give away, in certain restricted circumstances---usually I'll share these as well, since more people caring about my problem area is generally better for me.
Give the idea away for somebody else to work on, but continue contributing to the work in an advisory or other accessory manner.  This is the typical mode of operation for many professors, and can certainly be for a postdoc as well, even after you move on to another institution.  If you continue to contribute in a minor way to the work, you should receive minor credit (e.g., Nth authorship). 
Give the idea away for somebody else to work on, but don't contribute to the work beyond that.  This is great because you don't burden yourself with the extra work, but you should probably expect an acknowledgement at most, not authorship.


In short: in most cases, it's not the idea, but the work following the idea that counts.  We only think otherwise because we are scared or because our myths of science frequently mischaracterize the result of work as the idea.