Most common examples in the literature are rings of "higher virtues", having finite decompositions into irreducibles (maybe non-unique) -> atomic rings (Cohn), or Noether, or Prüfer, or GCD rings, or their intersections (UFD, Bézout, Dedekind and even more refined objects like PIDs, Eucl., fields). Why is it so much harder to find an easy "ID-only" ring?
2026-03-29 14:59:15.1774796355
What are some simple examples of "ID-only" rings? integral domains, that are neither atomic nor Noether nor Prüfer...
60 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in RING-THEORY
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- A commutative ring is prime if and only if it is a domain.
- Find gcd and invertible elements of a ring.
- Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain.
- Prove that $Z[i]/(5)$ is not a field. Check proof?
- If $P$ is a prime ideal of $R[x;\delta]$ such as $P\cap R=\{0\}$, is $P(Q[x;\delta])$ also prime?
- Let $R$ be a simple ring having a minimal left ideal $L$. Then every simple $R$-module is isomorphic to $L$.
- A quotient of a polynomial ring
- Does a ring isomorphism between two $F$-algebras must be a $F$-linear transformation
- Prove that a ring of fractions is a local ring
Related Questions in NOETHERIAN
- In a left noetherian ring, does having a left inverse for an element guarantee the existence of right inverse for that element?
- Prove that the field $k(x)$ of rational functions over $k$ in the variable $x$ is not a finitely generated $k$-algebra.
- Ascending chain of proper submodules in a module all whose proper submodules are Noetherian
- Noetherian local domain of dimension one
- Dimension of Quotient of Noetherian local ring
- Is $\mathbb{Z}[\frac{1}{2}]$ Noetherian?
- Finitely generated modules over noetherian rings
- Simplicity of Noetherian $B$, $A \subseteq B\subseteq C$, where $A$ and $C$ are simple Noetherian domains
- Why noetherian ring satisfies the maximal condition?
- If M is a a left module over $M_n(D)$ where $D$ is a division ring, M is Noetherian iff Artinian
Related Questions in INTEGRAL-DOMAIN
- Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain.
- GCD of common divisors in integral domain
- Jacobson radical of formal power series over an integral domain
- Characteristic of an integral domain: doubt in the proof
- When prime element in an integral domain stays prime in integral extension
- Localization in integral domains
- Module over integral domain, "Rank-nullity theorem", Exact Sequence
- Why must a domain be nonzero?
- Contraction of primes associated to nonzerodivisors
- Abstract algebra for $\mathbb Z_p[i]$ form a field
Related Questions in UNIQUE-FACTORIZATION-DOMAINS
- Extension and restriction of involutions
- Why is this element irreducible?
- What is the correct notion of unique factorization in a ring?
- A question about unique factorization domain
- Is the union of UFD an UFD?
- Is $F_p^{l}[t]$ is a UFD
- etymology of smoothness
- $2=(1+i)(1-i)$ what does that imply in $\mathbb{Z}[i]$?
- Is there a way of proving that $\mathbb{Z}[i]$ and $\mathbb{Z}[\sqrt{-2}]$ are UFD s without showing that they are euclidian domains?
- What implies that $D[X]$ is an UFD?
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?
Using this map at DaRT one would look for domains which are not N-1, not Goldman, not Mori, and not atomic. By avoiding these we dodge more stringent conditions too.
Interestingly, I did not realize there was no non-Mori domain confirmed yet, so thank you for asking.
I suppose we could additionally ask for a domain that isn't any of these four things, or at least combinations of them. But that gets complicated.
Of course, it is not really sensible to ask for an "ID-only ring" because the pool of conditions to avoid is potentially infinite. In general, every extra condition you have to dodge makes the problem that much harder. I think the best you could possibly hope to obtain is a domain that isn't the four things above, plus maybe some other conditions you are interested in that DaRT does not have.
Update:
It seems that I have been ignorant of another fact that will change the above map: Mori -> ACCP, so it actually is situated above atomic. I will try to adapt this in the near future, and also include "noetherian" in there.