I am reading Atiyah's Commutative Algebra chapter on Primary Decomposition. I understand the proofs but I have no intuition as to
- How did one come up with definition of a primary ideal $q$ as
$xy \in q \Rightarrow$ either $x \in q$ or $y^n \in q$ for some $n>0$
- Why we know that this decomposition is a "nice" one.
Perhaps there are other decompositions, that satisfies some uniqueness properties. What is so special/"canonical" about this hecomposition?
- What properties of primary decomposition are "important" - what are the consequences?
I am confused as to what properties yields more important results. I have skipped ahead and look at some corollaries; some results can also be proven with out the theory primary decomposition.
What I am look here is a more detailed motivation/intuition. Rather than "prime decomposition in the ideals". Feel free to close this post if it is too broad.