List questions from principal-ideal-domains

Published on 28 Sep 2021 - 14:06

149

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For which $d$, the algebraic integers of $K =\Bbb{Q}(\sqrt d)$ is a PID?

Published on 23 Feb 2026 - 16:26

126

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Writing modules over PIDs as direct sums of cyclics

Published on 19 Oct 2021 - 22:32

61

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Show that $\mathbb{Z}[i]=\{m+in\colon\; m,n\in\mathbb{Z}\}$ is a P.I.D

Published on 26 Mar 2026 - 17:32

281

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Hypothesis of Structure Theorem for Finitely-Generated Modules over PID's

Published on 28 Oct 2021 - 4:56

363

Views

Rank and Nullity is preserved by multiplication of invertible matrices (PID).

Published on 05 Nov 2021 - 13:26

102

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Confusion about multiplying two ideals.

Published on 05 Nov 2021 - 14:29

108

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Chinese Remainder Theorem for Hurwitz quaternions

Published on 01 Apr 2026 - 4:36

18

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Checking if given polynomials are units in $\mathbb{Z}_7[x]$

Published on 22 Nov 2021 - 1:03

50

Views

Write $(7,\omega -5)$ as principal ideal in $\mathbb{Z}[\omega]$

Published on 25 Nov 2021 - 16:47

112

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Rings of integers of number field and free modules over a PID

Published on 29 Mar 2026 - 19:28

181

Views

How $\frac{Z[x,y]}{\langle y+1\rangle}$ is Unique factorization domain?

Published on 30 Mar 2026 - 20:53

247

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Let $D$ be a PID. Show that $d\in D$ is a least common multiple of $a$ and $b$ iff $(a)\cap(b)=(d)$.

Published on 21 Dec 2021 - 22:36

30

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If $A$ is a Principal Ideal Domain, and $\mathfrak{a}$ its ideal. prove that $\frac{A}{\mathfrak{a}}$ is also a Principal Ideal Domain.

Published on 24 Dec 2021 - 8:43

377

Views

A prime ideal of a polynomial ring over a PID can be generated by two elements.

Published on 29 Dec 2021 - 20:40

List Question

why we can't write that $\pm2=2r(x) +xs(x)$ instead of $x=2v(x)$?

For which $d$, the algebraic integers of $K =\Bbb{Q}(\sqrt d)$ is a PID?

Writing modules over PIDs as direct sums of cyclics

Show that $\mathbb{Z}[i]=\{m+in\colon\; m,n\in\mathbb{Z}\}$ is a P.I.D

Hypothesis of Structure Theorem for Finitely-Generated Modules over PID's

Rank and Nullity is preserved by multiplication of invertible matrices (PID).

Confusion about multiplying two ideals.

Chinese Remainder Theorem for Hurwitz quaternions

Checking if given polynomials are units in $\mathbb{Z}_7[x]$

Write $(7,\omega -5)$ as principal ideal in $\mathbb{Z}[\omega]$

Rings of integers of number field and free modules over a PID

How $\frac{Z[x,y]}{\langle y+1\rangle}$ is Unique factorization domain?

Let $D$ be a PID. Show that $d\in D$ is a least common multiple of $a$ and $b$ iff $(a)\cap(b)=(d)$.

If $A$ is a Principal Ideal Domain, and $\mathfrak{a}$ its ideal. prove that $\frac{A}{\mathfrak{a}}$ is also a Principal Ideal Domain.

A prime ideal of a polynomial ring over a PID can be generated by two elements.

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