How can we explain the following step in the proof of Krull principal ideal theorem: $l\{ ((z):x^n)/(z) \}$ or $l\{ ((x^n):z)/(x^n) \}$ is finite?
$l(M)$ - length of module.
2025-01-13 07:56:37.1736754997
Question about proof of Krull principal ideal theorem
77 Views Asked by A.Shevchenko https://math.techqa.club/user/a-shevchenko/detail At
1
There are 1 best solutions below
Related Questions in COMMUTATIVE-ALGEBRA
- Relations among these polynomials
- Completion of a flat morphism
- Explain the explanation of the Rabinowitsch trick
- Does intersection of smooth divisors satisfy Serre $S_2$ criterion?
- Prime ideals $\mathfrak{p} \supset \mathfrak{a}$ are finite in one-dimensional Noetherian domain
- If module $M = N + mM$, then why is $m(M/N) = M/N$?
- Isomorphism of Localization of $A_\mathfrak{p}$ and $A_\mathfrak{q}$
- Question on an exercise in Atiyah-Macdonald.
- Tensor product of two fields
- Why the ideal of the union of coordinate axes in $\mathbb A^3$ can not be generated by two elements, i.e., is not a complete intersection?
Related Questions in MODULES
- Proper essential extensions
- If module $M = N + mM$, then why is $m(M/N) = M/N$?
- On the invariance of basis number
- Geometric intuition for coherent rings, modules, and sheaves
- Finitely generated iff direct limits of subobjects are bounded by subobjects
- Finding highest weight of a $gl(3)$ submodule of a $gl(4)$-module
- A simple Congruence Equation
- On the $R$-linear independence of free $R$-module
- Do all submodules of $R^n$ have a complement in $R^n$?
- Formulas give irreducible representation, $SL(2, \mathbb{C})$.
Related Questions in NOETHERIAN
- Prime ideals $\mathfrak{p} \supset \mathfrak{a}$ are finite in one-dimensional Noetherian domain
- Local Noetherian ring and its invariants
- A field is always noetherian
- Zerodivisors and grade of an ideal
- $A \oplus M$ is Noetherian iff $A$ is Noetherian and $M$ is a f.g. $A$-module
- What does this field notation mean?
- Direct sum of noetherian rings
- Localization of a one-dimensional noetherian integral domain
- Examples of Noetherian Domains of Dimension One
- Noetherian criterion and principal ideals.
Related Questions in LOCAL-RINGS
- Can a product of non-principal ideals be principal, in a local ring?
- Local Ring with an another definition
- About a short proof of Krull principal ideal theorem
- System of parameters in Noetherian local rings
- Reference on a result about integral closures.
- Commutative Algebra "mess": recover the complete local rings of the normalization
- Local ring of an affine curve $K$ at a point $p\in K$
- Local group rings
- Maximal ideal in a local artinian ring.
- Is the sheaf of differentials of a smooth morphism locally free?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity
If $I\subset R$ is an $M$-primary ideal then $R/I$ is artinian. In particular, every ideal $J/I$ if $R/I$ is an $R/I$-module of finite length, hence an $R$-module of finite length.