Hi. I'm preparing a thesis in commutative algebra, and when I say this to my friends they always ask me what are the applications to "real-world", and I don't know what to answer. This let me think that I'm studying something useless. I'm studying on the Matsumura and on the Herzog-Bruns. Any of you know some applications of this abstract algebra to the real-world?
[Math] Applications of commutative algebra
ac.commutative-algebraapplicationsapplied-mathematics
Related Solutions
For a reference on Cohen-Macaulay and Gorenstein rings, you can try "Cohen-Macaulay rings" by Bruns-Herzog.
Also, Huneke's lecture note "Hyman Bass and Ubiquity: Gorenstein Rings" is a great introduction to Gorenstein rings, very easy to read and to the point, I highly recommend it.
EDIT: Since this question is already bumped up, I will take this opportunity to make a longer list.
There are of course some classic references which are still very useful (I find myself having to look in them quite often despite the new sources available): Bourbaki, EGA IV, Serre's "Local Algebras" (very nice read and culminated in the beautiful Serre intersection formula).
There has been some work done in commutative algebra since the 60s, so here is a more up-to-date list of reference for some currently active topics (Disclaimer: I am not an expert in any of these, the list was formed by randomly looking at my bookself, and put in alphabetical order (-:). This is community-wiki, so feel free to add or edit or suggest things you found missing.
Cohen-Macaulay modules, from a representation theory perspective: Yoshino is excellent. Another one is being written.
Combinatorial commutative algebra: Miller-Sturmfels.
Free resolutions (over non-regular rings): Avramov lecture note
Geometry of syzygies: Eisenbud, shorter but free version here.
Homological conjectures: Hochster, Roberts (more connections to intersection theory), Hochster notes.
Integral closures: Huneke-Swanson, which is available free at the link.
Intersection theory done in a purely algebraic way: Flenner-O'Carrol-Vogel (for a very interesting story about this, see Eisenbud beautiful reminiscences, especially page 4)
Local Cohomology: Brodmann-Sharp, Huneke's lecture note (very easy to read), 24 hours of local cohomology (I have been told that this one was a pain to write, which is probably a good sign).
Tight closure and characteristic $p$ method: Huneke, Karen Smith's lecture note (more geometric, number 24 here), and of course many well-written introductions available on Hochster website.
By comparing the tables of contents, the two books seem to contain almost the same material, with similar organization, with perhaps the omission of the chapter on excellent rings from the first, but the second book is considerably more user friendly for learners. There are about the same number of pages but almost twice as many words per page. The first book was almost like a set of class lecture notes from Professor Matsumura's 1967 course at Brandeis. Compared to the second book, the first had few exercises, relatively few references, and a short index. Chapters often began with definitions instead of a summary of results. Numerous definitions and basic ring theoretic concepts were taken for granted that are defined and discussed in the second. E.g. the fact that a power series ring over a noetherian ring is also noetherian is stated in the first book and proved in the second. The freeness of any projective modules over a local ring is stated in book one, proved in the finite case, and proved in general in book two. Derived functors such as Ext and Tor are assumed in the first book, while there is an appendix reviewing them in the second. Possibly the second book benefited from the input of the translator Miles Reid, at least Matsumura says so, and the difference in ease of reading between the two books is noticeable. Some arguments in the second are changed and adapted from the well written book by Atiyah and Macdonald. More than one of Matsumura's former students from his course at Brandeis which gave rise to the first book, including me, themselves prefer the second one. Thus, while experts may prefer book one, for many people who are reading Hartshorne, and are also learning commutative algebra, I would suggest the second book may be preferable.
Edit: Note there are also two editions of the earlier book Commutative algebra, and apparently only the second edition (according to its preface) includes the appendix with Matsumura's theory of excellent rings.
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Best Answer
The book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by Cox, Little & O'Shea, contains some "real world" applications, specifically chapter 6 (of the 3rd edition) is titled "Robotics and Automatic Geometric Theorem Proving".