List questions from principal-ideal-domains

Published on 24 Feb 2014 - 0:08

444

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Nonprincipal prime ideals contain two relatively prime elements

Published on 29 Mar 2026 - 20:49

9.2k

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Is any UFD also a PID?

Published on 29 Mar 2026 - 20:49

925

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Homology out of Smith normal form: simultaneous or independent diagonalization?

Published on 04 Mar 2014 - 1:33

104

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Show $M(\mathfrak{p})$ is an $R$-submodule of $M$

Published on 04 Mar 2014 - 23:46

143

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$IJ =(I\cap J)(I+J)$ holds in a PID? $I,J$ Ideals of a PID

Published on 06 Mar 2014 - 1:38

8.9k

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Prove that $n^2+n+41$ is prime for $n<40$

Published on 10 Mar 2014 - 10:05

40

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for $n\geq 3$ $C(n,k)$for the following mapping

Published on 12 Apr 2017 - 23:48

12.3k

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Is every PID a UFD?

Published on 29 Mar 2026 - 20:47

116

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Existence of a invertible matrix with given first column

Published on 18 Apr 2017 - 13:47

1k

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Quotient of a PID by a prime ideal is a PID too.

Published on 19 Apr 2017 - 5:43

2.2k

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Show that if $D$ is a PID, then any two elements of $D$ have a g.c.d. which can be written in the form $xa+yb$, for some $x,y\in D$.

Published on 20 Apr 2017 - 0:03

42

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Suppose $R$ is a PID, $p ∈ R$ is a prime element, and $s ≥ 1$. Prove the $R$-module $R/p^s$ has no proper submodule which is a summand.

Published on 20 Apr 2017 - 6:42

37

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For $R=\mathbb{Z}[x]/ (x^2+5)$, find an ideal $I$ of $R$ which is not principal.

Published on 20 Apr 2017 - 20:19

896

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All $\mathbb{Z}$-submodules of $\mathbb{Z}\oplus\mathbb{Z}$ are free.

Published on 29 Mar 2026 - 4:47

List Question

Solution of a congruence equation in a PID

Nonprincipal prime ideals contain two relatively prime elements

Is any UFD also a PID?

Homology out of Smith normal form: simultaneous or independent diagonalization?

Show $M(\mathfrak{p})$ is an $R$-submodule of $M$

$IJ =(I\cap J)(I+J)$ holds in a PID? $I,J$ Ideals of a PID

Prove that $n^2+n+41$ is prime for $n<40$

for $n\geq 3$ $C(n,k)$for the following mapping

Is every PID a UFD?

Existence of a invertible matrix with given first column

Quotient of a PID by a prime ideal is a PID too.

Show that if $D$ is a PID, then any two elements of $D$ have a g.c.d. which can be written in the form $xa+yb$, for some $x,y\in D$.

Suppose $R$ is a PID, $p ∈ R$ is a prime element, and $s ≥ 1$. Prove the $R$-module $R/p^s$ has no proper submodule which is a summand.

For $R=\mathbb{Z}[x]/ (x^2+5)$, find an ideal $I$ of $R$ which is not principal.

All $\mathbb{Z}$-submodules of $\mathbb{Z}\oplus\mathbb{Z}$ are free.

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