List questions from unique-factorization-domains

Published on 16 Apr 2026 - 11:18

464

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"the prime factorization of an element, if it exists, is always unique"

Published on 14 Apr 2026 - 11:55

195

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Intersection of ideals $I=(2x)$ and $J=(2x^2)$ of $\mathbb{Z}[2x,2x^2,2x^3,\dots]$ is not finitely generated.

Published on 14 Apr 2026 - 20:13

29.8k

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Ring of integers is a PID but not a Euclidean domain

Published on 13 Apr 2026 - 1:30

81

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If an integral domain $R$ has a factorization basis, is it a UFD?

Published on 14 Apr 2026 - 3:39

101

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Are there infinitely many real quadratic number fields with unique factorization?

Published on 12 Apr 2026 - 23:33

3.7k

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How to show that $\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD

Published on 11 Apr 2026 - 9:12

1.3k

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Let $F$ be a field and let $R$ be the $F$-subalgebra of $F[x]$ generated by $x^2$ and $x^3.$ Show that $R$ is not a unique factorization domain

Published on 16 Apr 2026 - 2:57

122

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Multiplicative homomorphism of Euclidean domains: Do irreducibles never map to reducibles?

Published on 16 Apr 2026 - 0:09

58

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Decomposition of UFD

Published on 11 Apr 2026 - 13:58

266

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

Published on 11 Apr 2026 - 23:15

65

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If $Q$ is a prime ideal of $R[x]$ then $QF[x]\cap R[x]=Q$

Published on 18 Apr 2026 - 9:45

1.6k

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Does there exist a UFD having only finitely many irreducibles?

Published on 14 Apr 2026 - 5:05

108

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Factor rings of polynomial rings and unique factorization

Published on 14 Apr 2026 - 21:26

1k

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Is $\Bbb{R}[X,Y]/(X^2+Y^2)$ a UFD or Noetherian?

Published on 15 Apr 2026 - 21:12

List Question

Grade of non principal Prime ideals in Noetherian UFDs

"the prime factorization of an element, if it exists, is always unique"

Intersection of ideals $I=(2x)$ and $J=(2x^2)$ of $\mathbb{Z}[2x,2x^2,2x^3,\dots]$ is not finitely generated.

Ring of integers is a PID but not a Euclidean domain

If an integral domain $R$ has a factorization basis, is it a UFD?

Are there infinitely many real quadratic number fields with unique factorization?

How to show that $\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD

Let $F$ be a field and let $R$ be the $F$-subalgebra of $F[x]$ generated by $x^2$ and $x^3.$ Show that $R$ is not a unique factorization domain

Multiplicative homomorphism of Euclidean domains: Do irreducibles never map to reducibles?

Decomposition of UFD

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

If $Q$ is a prime ideal of $R[x]$ then $QF[x]\cap R[x]=Q$

Does there exist a UFD having only finitely many irreducibles?

Factor rings of polynomial rings and unique factorization

Is $\Bbb{R}[X,Y]/(X^2+Y^2)$ a UFD or Noetherian?

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