Talking points in Commutative Algebra, please

[A.Magnus]

< Mentor Note -- thread moved to HH from the technical math forums >[/color]

My final assignment in graduate algebra is to write an essay about the relationship among the subjects we have learned so far this semester:

(1) Module
(2) The Field of Fractions of an Integral Domain
(3) Integrality
(4) Integrality and Fields
(5) Prime Ideals
(6) The Krull Dimension
(7) Noetherian Modules
(8) Noetherian Integral Domain
(9) Dedekind Domain

I know they belong to the Commutative Algebra, but here is my problem: My present math maturity level is capable of seeing only the nuts and bolts of those subjects, but not the whole overall picture. I would appreciate if somebody out there gives me the ideas about the big picture, their inter-connection, or any online resources that I can turn to.

Thank you very much for your time and help.

 

[Stephen Tashi]

Thoughts of a non-expert:

From a literary point of view, i would be nice to have a grand unifying theme - like "The quest to break things up into simpler pieces". (meaning things like "generators" , "bases", "factors" ). Things described only by the definition of a module are problematic in that regard. Modifying the assumptions improves the prospects (at least according to the PDF http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/noetherianmod.pdf ). Perhaps "The quest to break things up into simpler pieces" can be expressed in category theory, but that might not be to your teacher's taste.

Another grand theme is "The quest to see a thing as a part of a bigger thing" ( like "covering groups" for things, "spans" of things, the thing generated by a thing. I don't know enough about modules to see how that applies to your topics.

 

[A.Magnus]

@Stephen Tashi : Thanks! Since I am supposed to write an essay about the unifying big picture of all these terms, since up to this moment I still did not see any unifying thread out of them except that they all belong to Commutative Ring, I think I am going to write an essay about the history of Commutative Algebra, at least I can show there is thread connecting some of the terms. Thanks again for your time.

 

[A.Magnus]

< Mentor Note -- thread moved to HH from the technical math forums >

My final assignment in graduate algebra is to write an essay about the relationship among the subjects we have learned so far this semester:

(1) Module
(2) The Field of Fractions of an Integral Domain
(3) Integrality
(4) Integrality and Fields
(5) Prime Ideals
(6) The Krull Dimension
(7) Noetherian Modules
(8) Noetherian Integral Domain
(9) Dedekind Domain

I know they belong to the Commutative Algebra, but here is my problem: My present math maturity level is capable of seeing only the nuts and bolts of those subjects, but not the whole overall picture. I would appreciate if somebody out there gives me the ideas about the big picture, their inter-connection, or any online resources that I can turn to.

Thank you very much for your time and help.

I am still struggling with this assignment. I had thought of writing a history of Commutative Ring, but it is not a good idea after a second though.

Let's narrow this question down like this: Instead of asking you on relationship among the above subjects, I am going to sharpen it into asking you on the relationship between any two or three of the above subjects. I think now it is now an easier question. Thank you again for your time and help.

 

[Stephen Tashi]

What is "integrality"?

 

[A.Magnus]

@Stephen Tashi : Here is the formal definition:

Let $S$ be a subring of $R$. An element $r$ of $R$ is called integral over $S$ if there exist elements $s_0, s_1, ..., s_{n-1}$ in $S$ with 1 $\le n$ and

$r^n + s_{n-1}r^{n-1} + ... + s_1r + s_0 = 0.$​

So basically, $r$ is the root of a monic polynomial, monic means the leading coefficient is 1. Here is one example of ring element which is integral over a subring: We have $(\sqrt2)^2 - 2 = 0$, thus the element $\sqrt2$ of $\mathbb R$ is integral over $\mathbb Z$. So, integrality is the noun of being integral over a ring.

Hope it helps and thank you very much for your time.