Exercises for commutative algebra

I’m currently studying for my final exam of commutative algebra. In the class we covered Artinian rings, Dedekind domains, Integral closures, Grobner basis, (disctrete )Valuation rings… I’m particulary interested in exercises involving the exactecness of $ \operatorname{Hom}(\cdot,N) $ and $ \operatorname{Hom}(N,\cdot) $ . Thought that you guys could provide some exercises. Thank you all in advance