This question was asked earlier on Mathstackexchange but was closed very very soon without any answer and then deleted by the system!

I am a PhD student ( 1st year) in a poor country with a corrupt academic system( very nepotistic, racist,open bullying and other serious issues). So, a lot of mathematics I used to study was through textbooks of Western authors( Springer, Wiley and others). Local textbooks are of poor quality. I had this question in mind for 4-5 years but I don't have any help of collegues as everything is a competition in the country and I think this question should be asked to many people and also because I don't have much guidance in real life. Journey to even get admission in a PhD program was very hard due to racism, lack of guidance and personal issues. My specialization is Commutative Algebra. Most things I learnt was through self study only.

Question: What approach should I have while studying advanced mathematical textbooks (PhD level and beyond)?

My Approach: I start self studying from page 1 of chapter 1. Remember at the definitions and also the intuition. I go through statement of theorems and the proofs and see if I can understand the logic in the proofs. If I can then I go towards corollaries and next theorems and in the end I try to do 50% of the exercises at the chapter end. Then same happens in chapter 2 till the end of the book.

After some time I don't remember the core ideas of the proof of theorems as I don't try to memorize the crux ideas of the proof but always make sure that I can understand how the proof works in the book but I can remember definitions, statements of theorems and some corollaries.

What are the pitfalls in this approach? Do I need to remember the core ideas of the proof of theorems?

I shall be very grateful for any answer!

This question was asked earlier on Mathstackexchange but was closed very very soon without any answer and then deleted by the system!

I am a PhD student (1st year) in a poor country with a corrupt academic system  (very nepotistic, racist,open bullying and other serious issues). So, a lot of mathematics I used to study was through textbooks of Western authors  (Springer, Wiley and others). Local textbooks are of poor quality. I had this question in mind for 4-5 years but I don't have any help of colleagues as everything is a competition in the country and I think this question should be asked to many people and also because I don't have much guidance in real life. Journey to even get admission in a PhD program was very hard due to racism, lack of guidance and personal issues. My specialization is Commutative Algebra. Most things I learnt was through self study only.

Question: What approach should I have while studying advanced mathematical textbooks (PhD level and beyond)?

My Approach: I start self studying from page 1 of chapter 1. Remember at the definitions and also the intuition. I go through statement of theorems and the proofs and see if I can understand the logic in the proofs. If I can then I go towards corollaries and next theorems and in the end I try to do 50% of the exercises at the chapter end. Then same happens in chapter 2 till the end of the book.

After some time I don't remember the core ideas of the proof of theorems as I don't try to memorize the crux ideas of the proof but always make sure that I can understand how the proof works in the book but I can remember definitions, statements of theorems and some corollaries.

What are the pitfalls in this approach? Do I need to remember the core ideas of the proof of theorems?

I shall be very grateful for any answer!