Articles of unique factorization domains

Show that in a UFD, prime elements are irreducible

Show that in a UFD, prime elements are irreducible Here is what I have so far: Suppose $p \in R$ is prime, then the ideal it generates is a prime ideal in $R$. If $ab$ (where neither $a$ nor $b$ are zero) belongs to $\langle p \rangle$ then we can say (without loss of generality) […]

$\mathbb{Z}$ is not a UFD

I am writing a short presentation (approximately 10 minutes long) on the topic of Fermat’s Last Theorem. Obviously, the details presented will have to be pretty sparse but I am compiling a handout as well. Part of this handout will include theorems and results that I will use in my presentation but do not have […]

Existence of the least common multiple in a Unique Factorization Domain

Let $x,y \in R$, where $R$ is a unique factorization domain. Let $P$ be the set of all representatives in each class of associate irreducible elements of $R$. Then, suppose $x,y\in R$ are nonzero, nonunit elements (First Question: How would this proof change if either $x$ or $y$ is zero or is a unit of […]

characterization of unique factorization domain

let $D$ be an integral domain from now on. In the lecture, my professor proves TFAE: 1. $D$ is a UFD 2. $D$ satisfies ACCP and for any $a$ in $D$ we have $a$ is prime $\iff$ $a$ is irreducible On the other hand, the GCD domain article in Wikipedia states that TFAE: 1. $D$ […]

$R$ is a unique factorization domain $\iff$ every prime minimal over a principal ideal is also principal

I’m trying to show that a ring $R$ is a unique factorization domain $\iff$ every prime minimal over a principal ideal is also principal. I think the idea is to use the principal ideal theorem of Krull, but I don’t know how to connect principal ideal properties with unique factorization domain properties. I know that […]

$R/\mathfrak p$ not always a UFD

I am looking for a nice counterexample that for a UFD $R$ and $\mathfrak p\subset R$ a prime ideal, $R/\mathfrak p$ is not always a UFD as well.

Possible Irreducible but NOT prime!

UFDs are integrally closed Using the main result proved in the link I want to show that: $ A $={$a+b\sqrt {2}$ |$ a \in \mathbb Z$ & $ b$ is an even integer} is a subring of $\mathbb Z[ \sqrt {2} ]$ BUT $A$ is NOT a UFD . My thought: I was trying to […]

Is $R$ a necessarily UFD for $X$ an infinite set of symbols?

Let $R$ denote a UFD, and let $X = \{x_0,\cdots,x_{n-1}\}$ denote a finite set of symbols. Then $R[X]$ is a UFD. This follows, since if $R$ is a UFD, then so too is $R[x],$ for any symbol $x \notin R$. Q. If $R$ is a UFD, and $X$ is an arbitrary set, possibly infinite, is […]

Why is F a UFD?

This question already has an answer here: Prove that any polynomial in $F[x]$ can be written in a unique manner as a product of irreducible polynomials in F[x]. 1 answer

How to show that $\Bbb Z/(xw-zy)$ is not a UFD

This question already has an answer here: Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain 1 answer