I really enjoyed the basic algebra course and wanted to teach myself a little more. So I am trying to learn commutative algebra from Atiyah-MacDonald and Eisenbud.
The department in our university is very good for analysis based subjects. I have enjoyed courses like basic analysis, measure theory and probability theory. I have a clean insight of what a theorem is trying to say, in these subjects, and I can make a mental picture before I formulate and prove propositions rigorously. In algebra, I dont seem to have this insight. I wish I could speak to people proficient in commutative algebra to get an insight and find the right style of thinking in this field. By the way, I have read this thread.
I will make the question specific:
1) I do not understand the idea of an ideal quotient. I know it's definition and I can prove the properties listed in Atiyah-MacDonald's book. But yet I have no idea of the big picture. What is it's purpose? How can I spot an ideal quotient?
2) I came across an idea called 'exact sequences' in Eisenbud's intro to modules. In two swift examples, he constructs exact sequences. I can verify that the second example is indeed an exact sequence. But I could not figure out how he constructed such an example!!
The second example was the exact sequence:
Given a ring $R$, an ideal $I \subset R$, $a \in R$
$0 \to \dfrac{R}{(I : a)} \to \dfrac{R}{I} \to \dfrac{R}{I + (a)} \to 0$
Is there an insight to this construction that, sort of, lets me guess the 'exact sequence' relation between the objects?
P.S: I will have more specific questions as I read along. Thank you for your answers.