Intereting Posts

“De-localization” of a Noetherian module?
$\int_{-2}^{2} \sin(x^5)e^{(x^8\sin(x^4))} dx$
CW complex structure on standard sphere identifying the south pole and north pole
Two 2d vector angle clockwise predicate
Why is the extension $k(x,\sqrt{1-x^2})/k$ purely transcendental?
Distribution of Ratio of Exponential and Gamma random variable
Interesting combinatorial identities
Maximum distance between points in a triangle
Question about the dirac $\delta$-function
What is the solution to the Dido isoperimetric problem when the length is longer than the half-circle?
What is Cramer's rule used for?
If a cyclic group has an element of infinite order, how many elements of finite order does it have?
Sheaves and complex analysis
Can two integer polynomials touch in an irrational point?
Borel Measures and Bounded Variation

Or in the contrapositive form

Is every one-dimensional UFD noetherian?

I know how to construct a non-noetherian UFD (polynomials in infinite number of variables over a field) and I know that it is even possible to construct a finite dimensional non-notherian UFD from here, but this example is 3-dimensional.

- In GCD domain every invertible ideal is principal
- A question about the equivalence relation on the localization of a ring.
- Example of Artinian module that is not Noetherian
- Showing that a certain map is not flat by explicit counterexample
- Spectrum of $\mathbb{Z}^\mathbb{N}$
- Proving normality of affine schemes

Since a noetherian one-dimensional UFD is a PID and vice versa, the question could be also rephrased as

Is every one-dimensional UFD a PID?

I guess the answer is negative, since it is a nice and elegant statement that have never heard about.

- Extension and contraction of ideals in polynomial rings
- $k$-algebra homomorphism of the polynomial ring $k$
- Infinite coproduct of rings
- Deduction of usual Cayley-Hamilton Theorem from “Determinant Trick”
- Connectedness of the spectrum of a tensor product.
- What are some examples of coolrings that cannot be expressed in the form $R$?
- Intersection of neighborhoods of 0. Subgroup?
- Proof of the single factor theorem over an arbitrary commutative ring
- Computing the quotient $\mathbb{Q}_p/(x^2 + 1)$
- The group of invertible fractional ideals of a Noetherian domain of dimension 1

Every one-dimensional UFD is a PID (hence noetherian).

Let $\mathfrak p$ be a non-zero prime ideal of $R$ and $a\in\mathfrak p$, $a\neq 0$. Then $a$ can be written as a product of prime elements, so $\mathfrak p$ contains a prime element, say $p$. Then $(p)\subseteq\mathfrak p$ and since the dimension of $R$ is $1$ we get $(p)=\mathfrak p$. This shows that all prime ideals of $R$ are principal, and therefore $R$ is a PID.

- Are complex determinants for matrices possible and if so, how can they be interpreted?
- Evaluation of $ \sum_{k=0}^n \cos k\theta $
- When is differentiating an equation valid?
- Solve system of 3 equations
- Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp $
- Fixed Points and Graphical Analysis
- Proving a combinatorics equality: $\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}$
- Homology of a simple chain complex
- Problem with the ring $R=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q &\Bbb Q\end{bmatrix}$ and its ideal $D=\begin{bmatrix}0&0\\ \Bbb Q & \Bbb Q\end{bmatrix}$
- Quadratic extensions in characteristic $2$
- pattern in decimal representation of powers of 5
- Difference Between Tensoring and Wedging.
- Spectral radius of the Volterra operator
- outer automorphisms of $S_6$
- What's the most efficient way to mow a lawn?