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13:12
@rschwieb can you suggest a path for someone who wants to gain as much knowledge as he can in ring theory, category theory especially non commutative rings. If he just passed MS and wants to do PhD in any of the above domain. I mean books, lectures, articles or anything to start with
 
1 hour later…
14:13
@Chaudhary I can only offer you advice from my narrow experience.

I think the most benefit is gained from having teachers and advisors engaged with you. I realize the selection of books is much wider than the selection of teachers. But I feel like it is worth saying the teachers played a bigger role than books for me.
When I was transitioning from masters to PhD I had a huge pile of books I consulted to tackle qualification exams. Having taken group and ring theory earlier, i knew that's where the material could be found. I had Dummit and Foote, Hungerford, Isaacs, Grillet's and Jain's books at that point, which I cross-referenced with each other when studying. Ultimately Isaacs and Jain's books served me well. Grillet's is still an excellent reference, too.
@Chaudhary During the beginning of the PhD though I had many more interesting books to read. Lam's two books, Anderson+Fuller, Goodearl's book on von Neumann regular rings, Grillet's book again, Jacobson's algeba books, kaplansky's books, they were all very helpful, and often contained perspectives that one doesn't find in the first batch of books i mentioned.
@Chaudhary Another piece of advice for your PhD: don't neglect learning some other branches while you are on the path to algebra. For me it was topology and analysis-type classes. At least you will get inspiration, and there's a chance you might discover you like something else better than algebra.
If I had to name a book that I have now that I wish I had when starting MS or PhD, it would have been Leinster's Basic Category Theory.
After you feel like you have a broad view (of course, it's impossible to have any sort of complete view at this point) of the literature, it's good to read classic and cutting edge papers on the sort of topics you are very interested in.
 
4 hours later…
18:50
Thank you very much @rschwieb your answers on stackexchange seems an art to me, inner me wants to become like you in mathematics, that’s why I asked you for suggestion. I like your above suggestion but probably good teacher is hard to find in my case (in my university) there is an automatically criterion to take students in PhD under your supervision, I do not have the freedom to chose my supervisor.
@Chaudhary Oh well, do the best with what you have I suppose! Luckily you can find a lot of interesting mathematicians online.

Don't aspire to be like me too hard if you want to be an academic mathematician, though. I did not have a strong taste for academia so I left long ago...
Glad to hear some of my answers have been of some use.

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