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I am curious if you know some good books that have problems well supplemented to "Commutative Algebra I-II" by Zariski/Samuel. I am really enjoying it, but it does not have any exercise, leaving me to try to come up with my own problems (it is fun to do, but I would like to solve some concrete problems too). Essentially, I would like books that contain a lot of problems about commutative algebra to supplement Zariski/Samuel (I am not planning to read those supplementary books).

Also, is it a good idea to read a book in the computational commutative algebra along with Zariski/Samuel? I am very interested in studying Grobner basis, which Z/S does not have.

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    I too am a fan of Zariski-Samuel. The book Matsumura, Commutative Ring theory is also a favorite, and has exercises. – 2017-02-18
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    @ReneSchipperus Are Matusumura's problems required reading of the book? I do not really want to do extra reading, just want to solve problems. – 2017-02-18

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Atiyah and Macdonald's book has a lot of exercises. The actual text part of the book is very concise, so even if there's stuff there that isn't in Zariski/Samuel, you won't have much you need to read. Just skip all the theorems whose statements are familiar to you and do the exercises.