# What is Commutative algebra: Definition and 21 Discussions

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers

Z

{\displaystyle \mathbb {Z} }
; and p-adic integers.Commutative algebra is the main technical tool in the local study of schemes.
The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.

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1. ### I Commutative algebra and differential geometry

In Miles Reid's book on commutative algebra, he says that, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. How does this work?
2. ### I Splitting ring of polynomials - why is this result unfindable?

Assume that ##P## is a polynomial over a commutative ring ##R##. Then there exists a ring ##\tilde R## extending ##R## where ##P## splits into linear factor (not necessarily uniquely). This theorem, whose proof is given below, is difficult to find in the literature (if someone know a source, it...
3. ### I Is a commutative A-algebra algebraic over A associative?

Let ##A## be a ring and ##B## be a commutative algebra over A· Suppose that ##B## is generated by algebraic elements ##\beta\in B## over ##A##, meaning that ##\beta## fulfils a relation of the form ##P(\beta)=0##, with ##P\in A[X]##. Is ##B## necessarily associative ? NOTE: As usual...
4. ### A Is the proof of these results correct?

Hello, Below are two results with their proof. Of course, there may be several ways to prove these results, but I just need some checking. Can someone check carefully if the math is OK ? (but very carefully, because if there is a failure, I will be murdered :-) ) ? thx. Claim 1: Let ##L/K## be...
5. ### Algebra Supplementary Problems for Zariski/Samuel's Commutative Alg.

Dear teachers, I am curious if you know some good books that have problems well supplemented to "Commutative Algebra I-II" by Zariski/Samuel. I am really enjoying it, but it does not have any exercise, leaving me to try coming up with my own problems (it is fun to do, but I would like to solve...
6. ### Best Books on Non commutative algebra.......

Dear friends! I am interested to study quantum physics at deep level.For that I think I should be well versed with many mathematical tools.Though I have some knowledge of algebra up to Galois Theory and some more related topics of commutative algebra. I am a mathematics teacher(more a student...
7. ### Maximal ideal not containing specific expression

May there exist an integral domain R, with fraction field K, that fulfills the following condition: there exists x\in K, x\not \in R and a maximal ideal \frak m of R{[}x{]}, such that \frak m does not contain x-a for any a\in R ? Motivation : I am trying to prove a difficult result. A way to...
8. ### When is integral closure generated by one element

Hello, This is not a homework problem, nor a textbook question. Please do not remove. Is there a concrete example of the following setup : R is an integrally closed domain, a is an integral element over R, S is the integral closure of R[a] in its fraction field, S is not of the form R{[}b{]} for...
9. ### Valuation ring - can someone explain this ?

Hello, I asked somebody a question, and didn't understood his answer. Can someone explain it to me ? My question was : Is there a valuation ring in ℚ(x,y), lying above the ideal <x,y> in the ring ℚ[x,y], whose residual field is a non-trivial extension of ℚ ? Here is his answer: This is not too...
10. ### Ramification group of valuations - need terminology

I am lost and need some terminology (also hopefully sources). Let L/K be a Galois extension, and w be a valuation of a L, lying above a valuation v of K. Notice that I do not suppose that w is discrete. Given α > 0 in the finite image of w, each of the following can easily been shown to be a...
11. ### I want to self teach College Algebra but I don't know how?

I was enrolled in College Algebra for this upcoming winter because the hours will overlap with the class that I enrolled for spring. The reason why I want to learn this, is to refresh my Algebra skills, since I forgot most of it (I only remember matrices, radicals and identities). The purpose...
12. ### Valuations and places - decomposition and inertia group

Hello, I feel very uncomfortable with some aspects of the theory of valuations, places, and valuation rings. Here is one of my problems : Assume that L/K is a finite Galois extension of fields, and that F is a place from K to its residual field k, whose associated valuation ring is discrete. F...
13. ### Talking points in Commutative Algebra, please

< 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)...
14. ### MHB Sums of Ideals - R. Y. Sharp "Steps in Commutative Algebra"

In R. Y. Sharp "Steps in Commutative Algebra", Section 2.23 on sums of ideals reads as follows: ------------------------------------------------------------------------------ 2.23 SUMS OF IDEALS. Let ( {I_{\lambda})}_{\lambda \in \Lambda} be a family of ideals of the commutative ring R . We...
15. ### MHB Polynomial Rings - Lemma 1.13 from Sharp: Steps in Commutative Algebra

I am reading R.Y. Sharp: Steps in Commutative Algebra. Lemma 1.13 on page 7 (see attachment) reads as follows: -------------------------------------------------------------------------------------- 1.13 LEMMA. let R be a commutative ring, and let X be an indeterminate; let T be a commutative...
16. ### Are Commutative Algebra and Algebraic Geometry useful for physics?

Are "Commutative Algebra" and "Algebraic Geometry" useful for physics? Hello, I'm considering taking Commutative Algebra, and perhaps even Algebraic Geometry (for which the previous is a prerequisite). In the first place I would take it for the enjoyment of mathematics and to give me an...
17. ### Momentum operator's relation to commutative algebra

how is the quantum momentum operator (being a linear differential op.) related to commutative algebra?
18. ### Solutions to a set of polynomials (Commutative Algebra)

Hi I have a set of nonlinear equations f_i(x_1,x_2,x_3...) and I want to find their solutions. After doing some reading I have come across commutative algebra. So to simplify my problem I have converted my nonlinear equations into a set of polynomials p_i(x_1,x_2,x_3...,y_1,y_2...) by...
19. ### Graded Commutative Algebra: A Comprehensive Reference

Is there a good reference for commutative algebra of graded rings and modules? I've only found little bits and pieces in other texts (e.g. Hartshorne's Algebraic Geometry), and I would like to avoid having to dive fully into the theory of modules over preadditive categories! (And I'd prefer...
20. ### Commutative Algebra: Study of Rings & Groups

Why is CA mostly to do with the study of rings? Why not study more commutative groups? Or are most group noncommutative?
21. ### Commutative Algebra & Geometric Group Theory for String Theory

How useful is it to study commutative algebra for the understanding and development of string theory? What about geometric group theory? Which is more useful for string theory and why?