List questions from unique-factorization-domains

Published on 14 Nov 2017 - 22:36

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Product of Primary Ideals is the Intersection of Prime Ideals

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Finite Number of Principal Ideals

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Show that every principal ideal domain is a unique factorisation domain

Published on 25 Nov 2017 - 14:00

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Let $R$ be a UFD. Show that if $bc = ax$, then $b \in (a)$ or $c \in (a)$

Published on 25 Nov 2017 - 16:56

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Defining irreducible polynomials over polynomial rings

Published on 28 Mar 2026 - 13:40

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Prove that any element in a ring A can be written as a product of irreducible but A is not a UFD

Published on 28 Nov 2017 - 19:01

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In a UFD if $x$ is irreducible, then $(x)$ is prime

Published on 28 Nov 2017 - 22:42

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A PID that is a sub ring of a Non-PID

Published on 29 Nov 2017 - 18:25

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Every prime ideal of height 1 in a UFD is principal

Published on 03 Dec 2017 - 1:39

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In a UFD in which each maximal ideal is principal, ideal generated by two relatively prime elements is whole ring

Published on 03 Dec 2017 - 5:34

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Prove that a UFD $R$ is a PID if every nonzero prime ideal in $R$ is maximal.

Published on 06 Dec 2017 - 10:04

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$\mathbb{R}$ is not the field of fractions of a UFD

Published on 12 Dec 2017 - 0:05

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Factorization of ideal of a quotient ring

Published on 13 Dec 2017 - 6:59

243

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Primitive roots in integer rings of number fields with class number 1

Published on 21 Dec 2017 - 18:13

List Question

Proper Ideal is a Product of Maximal Ideals

Product of Primary Ideals is the Intersection of Prime Ideals

Finite Number of Principal Ideals

Show that every principal ideal domain is a unique factorisation domain

Let $R$ be a UFD. Show that if $bc = ax$, then $b \in (a)$ or $c \in (a)$

Defining irreducible polynomials over polynomial rings

Prove that any element in a ring A can be written as a product of irreducible but A is not a UFD

In a UFD if $x$ is irreducible, then $(x)$ is prime

A PID that is a sub ring of a Non-PID

Every prime ideal of height 1 in a UFD is principal

In a UFD in which each maximal ideal is principal, ideal generated by two relatively prime elements is whole ring

Prove that a UFD $R$ is a PID if every nonzero prime ideal in $R$ is maximal.

$\mathbb{R}$ is not the field of fractions of a UFD

Factorization of ideal of a quotient ring

Primitive roots in integer rings of number fields with class number 1

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