List questions from principal-ideal-domains

Published on 26 Mar 2026 - 17:38

465

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$R$ be an infinite commutative ring with unity such that for every non-zero ideal $I$ , $R/I$ is finite ; then is $R$ a PID or at least Noetherian?

Published on 27 Mar 2026 - 20:21

174

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Find $\alpha \in \mathbb{Z}[i]$ such that $(3+5i,1+3i) = (\alpha)$

Published on 18 Feb 2016 - 22:33

277

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Show that I is a principal ideal

Published on 19 Feb 2016 - 17:34

482

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Prime ideal in R[x] is either principal or $\mathfrak p = (q,f)$

Published on 26 Feb 2016 - 2:43

69

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Is it possible to find $\alpha \in \mathbb{Z}[i]$ such that $(3+5i,1+3i)=(\alpha)$?

Published on 26 Feb 2016 - 19:43

5.3k

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Is $\mathbf{Z}[X]/(2,X^2+1)$ a field/PID?

Published on 02 Mar 2016 - 23:24

971

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$M$ is the (possibly infinite) direct sum of its $p$-primary components as $p$ runs over all primes of $R$.

Published on 03 Mar 2016 - 13:34

1.4k

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Applications of the Dedekind-Hasse criterion

Published on 05 Mar 2016 - 12:55

59

Views

Decomposition in a PID

Published on 08 Mar 2016 - 1:40

73

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Is $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID?

Published on 26 Mar 2026 - 7:48

137

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Is $\mathbb{Z}_p[\mathbb{Z}_p]$ a PID?

Published on 26 Mar 2026 - 7:50

3.5k

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Why is the polynomial ring of more than one variable not a PID?

Published on 12 Mar 2016 - 16:50

243

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Decomposition of Free Module over PID

Published on 12 Mar 2016 - 19:44

758

Views

Show that for a field $F$, the polynomial ring $F[x_1, x_2, \ldots, x_n]$ is not a PID for $n>1$.

Published on 13 Mar 2016 - 19:55

List Question

Torsion elements with relatively prime orders

$R$ be an infinite commutative ring with unity such that for every non-zero ideal $I$ , $R/I$ is finite ; then is $R$ a PID or at least Noetherian?

Find $\alpha \in \mathbb{Z}[i]$ such that $(3+5i,1+3i) = (\alpha)$

Show that I is a principal ideal

Prime ideal in R[x] is either principal or $\mathfrak p = (q,f)$

Is it possible to find $\alpha \in \mathbb{Z}[i]$ such that $(3+5i,1+3i)=(\alpha)$?

Is $\mathbf{Z}[X]/(2,X^2+1)$ a field/PID?

$M$ is the (possibly infinite) direct sum of its $p$-primary components as $p$ runs over all primes of $R$.

Applications of the Dedekind-Hasse criterion

Decomposition in a PID

Is $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID?

Is $\mathbb{Z}_p[\mathbb{Z}_p]$ a PID?

Why is the polynomial ring of more than one variable not a PID?

Decomposition of Free Module over PID

Show that for a field $F$, the polynomial ring $F[x_1, x_2, \ldots, x_n]$ is not a PID for $n>1$.

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