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This is my frist course in commutative algebra. I took two weeks ago Zariski topology which is interesting. , but my text book (Sharp) doesn't have this topic. I want to know how I can find spectrum for some ring like Z[x] and others. I found them as exercises in commutative algebra by Atiyah Macdonald. Some solutions are available online for this book, but I prefer to find book that has more information about spec R for some rings.

Any help will be appreciated

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    google.eisenbud's commutative algebra. for Spec Z[x] use google and search for "mumford's treasure map" = best possible explanation. (it is not that hard to determine Spec Z[x]). – 2017-02-07

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I think Vakil's FOAG develops it pretty well, though he leaves a decent amount of details to exercises. You could also try Bosch's book.

Honestly though my personal recommendation (feel free to ignore it) would just be to use Atiyah's book. He gives all the facts you should know on a first exposure; you just need to figure out the proofs for yourself, which is nice practice and should be relatively straightforward for the most part

Also a side note: finding Spec Z[x] explicitly is highly nontrivial

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    Thnaks, Alex. I am asking about Vakil's FOAG he has a book or notes. If so, give the book's name – 2017-02-07
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    That is the name. Look up "Vakil the Rising Sea" – 2017-02-07
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    @Team here is a link if you still haven't found it: http://math.stanford.edu/~vakil/216blog/FOAGdec2915public.pdf – 2017-02-07