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I took a course in abstract algebra (till galois theory), topology (with some very basic algebraic topology), smooth manifolds, complex analysis and never did I struggle even epsilon close to how I am now struggling to learn commutative algebra.

The main problem is that unlike other topics, I literally have zero intuition, and I can't prove a single theorem in the text by myself without reading the proof (I don't have the tiniest of clue on which direction to proceed). On top that my memory isn't particularly good, today I was not able to reconstruct the proof of Artinian ring implies Noetherian ring (even though I read the proof like a few weeks ago).

Today I was reading about Krull-Akizuki theorem (that if $A$ is integral, and suppose $A \subset B \subset F(A)$ be chain of inclusion of subrings. Then if $A$ is noetherian with every nonzero prime ideal maximal, then so is $B$). I tried to prove it myself, figured the condition implies $A/aA$ is artinian, and the conclusion is equivalent to saying $B/bB$ is artinian too for any $b \neq 0 \in B$ and then I have no tiniest clue on how to proceed from here. I read the proof, but I learnt nothing. Or that there was the section on primary decomposition, I again had literally zero clue about what's going on. I don't have any idea on when to localize, when to use primary decomposition or when to use filtrations etc except trying completely random things wiht the hope that they will work out

How to have intuition ? Or even if I throw intuition out of the window, how do I get enough ``getting-used-to" with this commutative algebra business so that I can solve the problems without much thinking (I can do this for example for most baby Rudin/Dummit-Foote problems) ?

PS: I know that learning algebraic geometry would help (and that's what I'll do after I learn this properly), but this extends to a broader question of learning (abstract) stuff which is not immediately obvious (I'm facing similar problems when learning homology also), so a general answer will also be appreciated.

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    $\begingroup$ I think you maybe expect too much. Most people would not be able to reconstruct a proof that they had just read "a few weeks ago". To learn and understand proofs, you (or at least, most people) need to go over them many times over a period of weeks, or months. Read the proof, writing down the steps yourself as you along. Then try to write it again immediately, from memory. If you can't do that, go back to step 1. If you can, then try to do it again after an hour. Then the next day. Then three days after that, etc. You can't just read a proof and expect to remember it. $\endgroup$ – Qwertiops Jun 2 at 19:52
  • $\begingroup$ I often find it helpful to find several books which cover the same topic and try to compare and contrast the way they discuss the same topics. There are several questions on MSE about good intro commutative algebra references, here is one: math.stackexchange.com/questions/37364/… $\endgroup$ – walkar Jun 3 at 0:10
  • $\begingroup$ This comment is for sure not helpful, nevertheless its my long-term observation: mathematical ability frequently appears with a bias either towards analysis or towards algebra. $\endgroup$ – Hagen Knaf Jun 4 at 6:24

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