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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Can a unique factorisation domain have a largest prime?
Suppose $R$ is a UFD and $(R,\leq)$ is an ordered ring. Is it possible that $R$ has a largest prime element? Below is my attempt so far to answer this myself, though I'm still unsure what the ...
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Irreducible elements, prime elements, prime ideals, and maximal ideals [duplicate]
I'm trying to get the four concepts listed in the title straight in my mind and elucidate the relationship between them. Could someone check the following statements and let me know if they are all ...
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Mistake in Proof "Every unique factorization domain is a principal ideal domain"
While doing my Algebra HW, I "proved" that every unique factorization domain (UFD) is a principal ideal domain (PID). I know that this is not true, however I fail to see where exactly is ...
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What property a ring extension of UFDs $R\subseteq S$ must have, such that for $a,b\in R$, $a | b$ in $S \implies a | b$ in $R$?
Let $R,S$ be two UFDs, and $R \subseteq S$, a ring extension with the following property (P) $$(\forall a,b\in R) (\ a|b \ in \ S \longrightarrow a|b \ in \ R).$$
Question: What are some conditions ...
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R is UFD. R is PID if every prime ideal is principal. [duplicate]
Suppose not. We consider the set of all non-principal ideals, $S$. Order $S$ by inclusion. We show S satisfies all the conditions in Zorn's Lemma. So it has a maximal element. If we show the maximal ...
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normalization of an algebraic curve
I am trying to solve this exercise:
Let $f(x)$ be a polynomial with distinct roots different from $0$. What is the normalization of the algebraic curve $y^2=x^2f(x)$?
My attempt:
We call the base ...
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Are there rings where factorization is unique, but does not necessarily exist?
It feels like this should be a well-known question, but I can't find any related questions on this site by searching; apologies in advance if this is a duplicate.
I assume rings are commutative with ...
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Explicit proof of the fact that a domain which is not a UFD is not a PID
In the same spirit as this question, I would like to prove explicitely that if $R$ is a domain which is not a UFD, then it is not a PID.
I am interested in the case where there is an element $a\in R$ ...
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Need help understanding the proof showing the existence of gcd in an UFD.
Background:
Definition: Let $a_1,a_2,\ldots a_n$ be elements (not all zero) of an integral doain $R.$ A $\textbf{greatest common divisor}$ of $a_1,a_2,\ldots, a_n$ is an element of $d$ of $R$ such ...
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Need help understanding the proof that if $R$ is a PID, every nonzero, nonunit elements of $R$ is a product of irreducibles.
Background:
Lemma 10.9: Let $a$ and $b$ be elements of an integral domain $R.$ Then
$(a)\subset (b)$ if and only if $b\mid a.$
$(a)=(b)$ if and only if $b\mid a$ and $a\mid b.$
$(a)\subsetneq (b)$...
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Explicit proof of the fact that a non integrally closed domain is not a UFD
Context. Let $R$ be a domain. It is well-known that $R$ is a PID $\Rightarrow$ $R$ is a UFD $\Rightarrow$ $R$ is integrally closed (in its field of fractions).
In other words, we have $R$ is not ...
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Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$
How do I factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$?
I checked that the discriminant $D = 16 -4 = 12$ is not a square ($12^{14} = -1 \mod 29$) so this polynomial has no roots. Therefore it's ...
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Factorization into irreducibles in $\mathbb{Z}[i\sqrt{n}]$ with square-free $n \in \mathbb{Z}^{+}$
I'm trying to arrive at a proof that every nonzero $\alpha \in \mathbb{Z}[i\sqrt{n}]$ ($n \in \mathbb{Z}^{+}$ square-free) that is not an unit can be factored into the product of irreducible elements ...
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Unique factorization domain property in $\mathbb{Z[\mathcal{i}]}$
Acknowledging that $\mathbb{Z[\mathcal{i}]}$ is a UFD, show that $11 + 14i, 23 − 12i, −91 + 230i, 305 + 192i$ cannot all be prime.
By direct calculation, I checked that $$(11+14i)(305+192i)=(23-12i)(-...
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Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$ then $R_S$ is also UFD
Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$ then $R_S$ is also UFD.
There are several answer on M.SE like here, here. But none of them resolve the ...
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