List questions from principal-ideal-domains

Published on 02 Sep 2019 - 17:38

601

Views

Every finitely generated module of a PID is a direct sum of a free module and of a torsion module

Published on 25 Mar 2026 - 13:34

796

Views

$\mathbb{Z}[\sqrt{-7}]$ is not an Euclidian Domain

Published on 25 Mar 2026 - 12:26

65

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For a PID $R,$ what is the rank of the submodule given by $S=\{(x_1,\ldots,x_n) \in R^n: x_1 +\cdots +x_n=0\}$?

Published on 25 Mar 2026 - 15:12

351

Views

The ideal which is not principal in $\Bbb Z[i\sqrt{6}]$

Published on 25 Mar 2026 - 12:27

56

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If $R$ is a PID and $M$ is finitely generated $R$-module, why is $M=\operatorname{coker}[R^m\to R^n]$

Published on 25 Mar 2026 - 18:49

65

Views

Being $f,g\in k[x,y]$ irreducible polynomials, there is $a,b\in k[x,y]$ and $c(x)\neq 0$ s.t. $af+bg=c(x)$

Published on 27 Mar 2026 - 14:52

195

Views

Torsionfree modules over PIDs

Published on 21 Sep 2019 - 21:08

195

Views

Dimension of free submodule of a free module over a PID

Published on 22 Sep 2019 - 14:00

173

Views

Projective Modules on PIDs

Published on 22 Mar 2014 - 19:16

611

Views

Structure Theorem For PIDs

Published on 23 Mar 2014 - 19:25

287

Views

Abstract Algebra: integral domain and principal ideal domain

Published on 25 Mar 2026 - 13:55

10.1k

Views

Proof for maximal ideals in $\mathbb{Z}[x]$

Published on 05 Apr 2014 - 15:17

447

Views

Proof for Unique Factorization Domain

Published on 25 Mar 2026 - 15:45

1.5k

Views

Finitely generated graded modules over $K[x]$

Published on 25 Mar 2026 - 7:40

List Question

Generator for ideal

Every finitely generated module of a PID is a direct sum of a free module and of a torsion module

$\mathbb{Z}[\sqrt{-7}]$ is not an Euclidian Domain

For a PID $R,$ what is the rank of the submodule given by $S=\{(x_1,\ldots,x_n) \in R^n: x_1 +\cdots +x_n=0\}$?

The ideal which is not principal in $\Bbb Z[i\sqrt{6}]$

If $R$ is a PID and $M$ is finitely generated $R$-module, why is $M=\operatorname{coker}[R^m\to R^n]$

Being $f,g\in k[x,y]$ irreducible polynomials, there is $a,b\in k[x,y]$ and $c(x)\neq 0$ s.t. $af+bg=c(x)$

Torsionfree modules over PIDs

Dimension of free submodule of a free module over a PID

Projective Modules on PIDs

Structure Theorem For PIDs

Abstract Algebra: integral domain and principal ideal domain

Proof for maximal ideals in $\mathbb{Z}[x]$

Proof for Unique Factorization Domain

Finitely generated graded modules over $K[x]$

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