# Questions tagged [unique-factorization-domains]

A commutative ring with unity in which every nonzero, nonunit element can be written as a product of irreducible elements, and where such product is unique up to ordering and associates. Also called UFD or "factorial ring"

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### More examples that are elementary of GCD domains that are not UFD domains.

What are some more examples that are elementary of GCD domains that are not unique factorization domains? For my personal notes, I'm looking for examples of rings that belong to one rung but not the ...
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### Unique Factoring in $\mathbb{Z}[x]$

While in class, my professor gave us a proposition that if $R$ is a unique factoring domain, then $R[x]$ is also a unique factoring domain. While in class I didn't think too much about it, but now I ...
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### Proving that fractions can be written in lowest terms in a unique factorization domain [duplicate]

When proving that rational numbers can be written in their lowest terms, we used the well-ordering principle to choose a numerator that is the smallest and then it contradicts the fact that this ...
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### Linear systems over UFDs

When Hermite and Smith normal form are discussed in the textbooks (e.g. the one of Adkins and Weintraub) they assume the ring to be a PID. What goes wrong over UFDs? Can one still do something similar ...
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### Viewing domain elements as fractions via natural map into fraction field (abuse of notation)

We have a simple proposition saying that Proposition: Suppose $R$ is a UFD and let $Q(R)$ be the fraction field of $R$. Let $f(x)=\sum_{i=0}^n a_i x_i \in R[x]$ be a polynomial of degree $n\ge 1$. ...
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### How to prove $x^p+y^p=\displaystyle{\prod_{k=0}^{p-1} (x+\xi_p^ky)}$

I am reading a "proof" of Fermat's last theorem which uses the assumption that $\mathbb{Z}[\xi_p]$ is a UFD (which is actually not true in general). This proof uses the following ...
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### Problem with understanding why $\mathbb{C}[x, y] / (x^2 + y^2 - 1)$ is UFD

Reading this questions Ring of trigonometric functions with real coefficients we can see that $\mathbb{C}[x, y] / (x^2 + y^2 - 1)$ is a UFD because it's isomorphic to Laurent polynomial ring which is ...
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### Why is a semilocal domain $A$ a UFD, if its localisations at maximal ideals $A_{\mathfrak m}$ are UFDs?

This is Exercise 20.2 in Matsumura's Commutative Ring Theory: Let $A$ be an integral domain. We say that $A$ is locally UFD if $A_{\mathfrak m}$ is a UFD for every maximal ideal $\mathfrak m$. If $A$ ...
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### Is there are two prime elements which generate a proper ideal? [duplicate]

Let $A$ be a UFD, we know $A$ is a PID iff $A$ is a Bezout domain. Now I want to consider the case $A$ is not a PID. So the question is that does every UFD $A$ which is not PID admit two distinct ...
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### Example of integral domain which is not a UFD [duplicate]

I was looking for an integral domain which is not a UFD and I've thought to $\mathbb{C}[x,y]/(x-y^2)$ My questions are: Is it right? Is there any easier example?
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### D.C.C condition of principal ideals on UFD

The question is: Let $R$ be a UFD, and $I \neq (0)$ be an ideal of $R$. Prove that every descending chain of principal ideals containing $I$ must stabilize. Since for an UFD, A.C.C holds for principal ...
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### Number of ring structures on $\mathbb{Z}$ with the usual multiplication

The usual addition is not the only one that makes $\mathbb{Z}$ into a ring with the usual multiplication. Indeed, for any positive integer $n$, $\mathbb{Z}[x_1,x_2,...,x_n]$ is a unique factorization ...
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### Is $\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2+1)$ a unique factorization domain?

Let $\mathbb R$ be the field of real numbers. From Nagata's criterion for unique factorization domains, it follows that $\frac{\mathbb R[X_1,\ldots,X_n]}{(X_1^2+\ldots+X_n^2)}$ is a unique ...
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### Is it true that an uniformizing element of a discrete valuation ring $R$ is either zero or prime?

Let $R$ be a DVR. This means that it is a PID with a unique maximal ideal. Let us denote $m$ to be the maximal ideal. Then there is a unique irreducible element $\pi$ up to multiplication my a unit ...
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### Problem in understanding of If $R$ is a UFD, then $R[x]$ is also a UFD(Gauss's theorem)

Gauss's Theorem:If R is a UFD then so is R[X], the ring of polynomials over R. I am studying the the from this pdf. But I have problem in understanding. Proof: Take a (non-zero) polynomial $f ∈ R[X]$....
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### Minimal non-zero prime ideal in a UFD

I'm trying to understand the following Proposition: Proposition: Let $R$ be a UFD (unique factorization domain) with subset $P$ of the set of irreducible elements of $R$. Then $P$ is in bijective ...
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### Finding $x$ from $Ax = b$ with $QR$ factorization

I'm currently trying to find $x$ such as $A x = b$ where $A \in \mathbb{R}^{5\times 5}$, $x \in \mathbb{R}^{5}$ and $b \in \mathbb{R}^{5}$. I used for that the $QR$ factorisation. I know I have the ...
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### Is $\mathbb{Z}[\sqrt{2}]$ a UFD? [duplicate]

Is the quotient ring $\mathbb{Z}[X]/(X^3-2)$ a UFD? I don’t have any idea to find whether some ring is a UFD or not.
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### How can an irreducible polynomial be represented as a product of irreducible polynomials? [duplicate]

In Charles Pinter's Abstract Algebra, he presents the factorization theorem for polynomials. That is, Every polynomial $a(x)$ of positive degree in $F[x]$ can be written as a product \begin{equation*}...
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### Show that the ring $\mathbb C[x^3,x^2y,xy^2,y^3]$ is normal, but it is not a unique factorisation domain

OExercise I am trying to solve: Show that the ring $R = \mathbb{C}[x^{3}, x^{2}y, xy^{2}, y^{3}]$ is normal, but $R$ is not a unique factorisation domain. Normal means that $R$ is integrally closed ...
### Let $R:=\mathbb{Z}[\sqrt{-6}]$. Prove that $R$ is not a Unique Factorization Domain
Question: Let $R:=\mathbb{Z}[\sqrt{-6}]$. (a) Prove that $R$ is not a Unique Factorization Domain (b) Find an irreducible element of $R$ that is not prime (c) Find a non-principal ideal in $R$ (d) ...
### What are the integers that lack unique factorization in $\mathbb Z[\zeta_n],$ $n= 23$? [closed]
In the cyclotomic integers $\mathbb{Z}[\zeta_n]$, $n=23$ has class number $3$ and so unique factorization fails (https://en.wikipedia.org/wiki/Cyclotomic_field ). Can anyone give me examples of such ...