Let $(R, \mathfrak m) $ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
2026-03-24 23:46:44.1774396004
Valuation ring whose unique maximal ideal and every ideal of finite height is principal
133 Views Asked by user495643 https://math.techqa.club/user/user495643/detail AtRelated Questions in COMMUTATIVE-ALGEBRA
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- Extending a linear action to monomials of higher degree
- Tensor product commutes with infinite products
- Example of simple modules
- Describe explicitly a minimal free resolution
- Ideals of $k[[x,y]]$
- $k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements
- There is no ring map $\mathbb C[x] \to \mathbb C[x]$ swapping the prime ideals $(x-1)$ and $(x)$
- Inclusions in tensor products
- Principal Ideal Ring which is not Integral
Related Questions in MAXIMAL-AND-PRIME-IDEALS
- Prime Ideals in Subrings
- If $P$ is a prime ideal of $R[x;\delta]$ such as $P\cap R=\{0\}$, is $P(Q[x;\delta])$ also prime?
- Prime ideals of $\Bbb C[X, Y]$.
- The radical of the algebra $ A = T_n(F)$ is $N$, the set of all strictly upper triangular matrices.
- Primary decomposition in a finite algebra
- Spectrum of $\mathbb{Z}[\frac{1}{6}]$
- Does $\mathbb Z/{2}\times\mathbb Z/{2}$ have no maximal and prime ideal?
- characterizing commutative rings, with nilpotent nilradical , satisfying a.c.c. on radical ideals
- Maximal and prime ideal in an artinian ring
- ring satisfying a.c.c. on radical ideals, with nilpotent nilradical and every prime ideal maximal
Related Questions in VALUATION-THEORY
- Does every valuation ring arise out of a valuation?
- Calculating the residue field of $\mathbb{F}_q(t)$ with respect to the valuation $-\deg$
- Different discrete valuation rings inside the same fraction field
- Extensions of maximally complete fields
- State Price Density Integration - Related to Ito's lemma
- Valuation on the field of rational functions
- Is every algebraically closed subfield of $\mathbb C[[X]]$ contained in $\mathbb C$?
- Working out an inequality in the proof of Ostrowski's Theorem
- The image of a valuation is dense in $\mathbb{R}$
- Kernel of the map $D^2+aD+b : k[[X]] \to k[[X]]$ , where $D :k[[X]] \to k[[X]]$ is the usual derivative map
Related Questions in KRULL-DIMENSION
- Krull dimension of a direct product of rings
- Dimension of Quotient of Noetherian local ring
- Motivation behind the Krull Dimension of a ring
- ring satisfying a.c.c. on radical ideals, with nilpotent nilradical and every prime ideal maximal
- Why does a zero-dimensional irreducible space have no non-trivial open subsets?
- torsion-free modules $M$ over Noetherian domain of dimension $1$ for which $l(M/aM) \le (\dim_K K \otimes_R M) \cdot l(R/aR), \forall 0 \ne a \in R$
- Localization of PID, if DVR, is a localization at a prime ideal
- How to prove that $\dim \mathrm{Spec}~A = \dim \mathrm{Spec}~ A_\mathfrak{p} + \dim \mathrm{Spec}~A/\mathfrak{p}$.
- Proof of Krull's Height Theorem for irreducible affine varieties
- Dimension of certain type of finitely generated $k$-algebra
Related Questions in LOCAL-RINGS
- Noetherian local domain of dimension one
- Hom and tensor in local Ring
- Injective map from a module of finite rank to free module
- Rank of completion of a module
- Given a prime ideal $P$ in a valuation ring $A$, there is a valuation ring $B$ containing $A$ such that $B/PB$ is the fraction field of $A/P$ ?
- Commutative Noetherian, local, reduced ring has only one minimal prime ideal?
- Galois Theory for Finite Local Commutative Rings
- Complete rings with respect to an I-adic topology
- Statement of Lech's lemma
- Difference between two localizations
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- 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?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- 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
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?