Questions tagged [unique-factorization-domains]
The unique-factorization-domains tag has no usage guidance.
6
questions
19
votes
6
answers
2k
views
Nonfree projective module over a regular UFD?
What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...
8
votes
1
answer
1k
views
First-order UFD (factorial ring) condition / pre-Schreier rings
All rings in this post are commutative and with $1$.
Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...
7
votes
2
answers
572
views
Zermelo's proof for unique factorisation
In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof ...
5
votes
1
answer
247
views
Is every matrix involution over a UFD diagonalisable?
Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$).
Is every involution in $\mathrm{GL}_n(A)$ diagonalisable?
This is of ...
2
votes
0
answers
99
views
Characterizing all simple algebraic ring extensions of $\mathbb{C}[x]$ having no prime elements
Let $w$ be an algebraic element over $\mathbb{C}[x]$, with minimal polynomial $f(t)=c_mt^m+\cdots+c_1t+c_0$, $c_i \in \mathbb{C}[x]$.
Is it possible to characterize (in terms of the $c_j$'s) all ...
0
votes
0
answers
278
views
For which monic irreducible $f \in \mathbb{C}[x,y][T]$, $\mathbb{C}[x,y][T]/(f)$ is a UFD?
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic irreducible polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$, $0 \leq j \leq n-1$.
Denote $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}...