Insight of some concepts in commutative algebra 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.
 A: To answer your first question: I never really got the feel for ideal quotients until I realised it can be used to prove the following two facts: 


*

*The set of zero divisors in a commutative ring $A$ is a union of prime ideals. The way I proved this at first was using Krull's Lemma, but you can also use ideal quotients as in here:

*A ring in which every prime ideal is finitely generated is Noetherian.
If you would like general advice for commutative algebra, I am not an expert but here's what I can say. When I first started commutative algebra I had 0 intuition. One thing I realised that was helpful as I was going along was to draw a commutative diagram. For example, you should be aware that if $A \subset B$ is a finite ring extension, then given any $Q \subset A$ a prime ideal there is always a prime ideal $P \subset B$ lying over $A$. If you draw a commutative diagram, the proof of this fact is not hard. Also, since you mentioned Eisenbud, I remember proving some isomorphism in there using just a big diagram chase. So, draw a diagram if you can!
A: 2) contains the ideal quotient $(I:a)$, it really comes from multiplication with an element in the ring:
$$
\mu_a : R \rightarrow R \qquad r \mapsto ar
$$
then you compose with the projection of an arbitrary ideal $I$
$$
\mu_a : R \rightarrow R/I \qquad r \mapsto [ar]=ar+I
$$
this has the image $(a)+I/I$ in $R/I$, and the kernel is by definition $(I:a)$. If you use the isomorphism theorems you immediately get the exact sequence, which is a short way to state a lot of information
