List questions from principal-ideal-domains

Published on 24 Mar 2026 - 17:47

188

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A variation of the argument to prove that $\{m/n:n \text{ is odd },n,m \in \mathbb{Z}\}$ is a PID

Published on 24 Mar 2026 - 11:42

72

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Why is this element irreducible?

Published on 20 Mar 2026 - 2:22

184

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Quotient of normal ring by principal ideal

Published on 22 Mar 2026 - 5:13

189

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$R/(a) \oplus R/(b) \cong R/\gcd(a,b) \oplus R/\operatorname{lcm}(a,b) $

Published on 20 Mar 2026 - 3:16

2.7k

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Proving a prime ideal is maximal in a PID

Published on 20 Mar 2026 - 2:47

495

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Localization of PID, if DVR, is a localization at a prime ideal

Published on 20 Mar 2026 - 6:05

325

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Structure theorem for modules implies Smith Normal Form

Published on 20 Mar 2026 - 8:04

882

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Let R be a PID, B a torsion R module and p a prime in R. Prove that if $pb=0$ for some non zero b in B, then $\text{Ann}(B)$ is a subset of (p)

Published on 20 Mar 2026 - 8:37

58

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Why can't a finitely generated module over a PID be generated by fewer elements than the number of invariant factors?

Published on 20 Mar 2026 - 3:52

121

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Localization of Modules over Dedekind domains

Published on 20 Mar 2026 - 9:30

46

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Simultaneous bases for free modules over PIDs?

Published on 24 Mar 2026 - 23:59

54

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The ideals of $\mathbb{Z}_{(p)}$ = {$m/n \in \mathbb{Q} : p$ does not divide $n$}.

Published on 24 Mar 2026 - 23:46

33

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Show that $M = M_a \oplus M_b$.

Published on 17 Mar 2026 - 0:28

36

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Consider the ideal $I = \langle(3,4) \rangle$ of the ring $\mathbb{Z} ×\mathbb{Z}$. Prove that $(\mathbb{Z}×\mathbb{Z})/I$ is not a domain.

Published on 05 Apr 2018 - 18:28

List Question

Principal Ideal Ring which is not Integral

A variation of the argument to prove that $\{m/n:n \text{ is odd },n,m \in \mathbb{Z}\}$ is a PID

Why is this element irreducible?

Quotient of normal ring by principal ideal

$R/(a) \oplus R/(b) \cong R/\gcd(a,b) \oplus R/\operatorname{lcm}(a,b) $

Proving a prime ideal is maximal in a PID

Localization of PID, if DVR, is a localization at a prime ideal

Structure theorem for modules implies Smith Normal Form

Let R be a PID, B a torsion R module and p a prime in R. Prove that if $pb=0$ for some non zero b in B, then $\text{Ann}(B)$ is a subset of (p)

Why can't a finitely generated module over a PID be generated by fewer elements than the number of invariant factors?

Localization of Modules over Dedekind domains

Simultaneous bases for free modules over PIDs?

The ideals of $\mathbb{Z}_{(p)}$ = {$m/n \in \mathbb{Q} : p$ does not divide $n$}.

Show that $M = M_a \oplus M_b$.

Consider the ideal $I = \langle(3,4) \rangle$ of the ring $\mathbb{Z} ×\mathbb{Z}$. Prove that $(\mathbb{Z}×\mathbb{Z})/I$ is not a domain.

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