List questions from principal-ideal-domains

Published on 28 Oct 2018 - 19:00

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PID controller for 2 DOF system

Published on 26 Mar 2026 - 16:03

1.2k

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Let $D$ be a PID and $a$ and $b$ be nonzero elements of $D$. Prove that there exist elements $s$ and $t$ in $D$ such that $\gcd(a, b) = as + bt$.

Published on 26 Mar 2026 - 8:23

93

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non-PID which where all prime ideals are maximal

Published on 31 Oct 2018 - 20:50

328

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Proving Ascending Ideals Stabilize in PID

Published on 25 Mar 2026 - 7:40

69

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Prove that for distinct prime ideals $P, Q$ of PID $R$, $P^n + Q^m = R$.

Published on 06 Nov 2018 - 11:18

42

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Factorization Domain in which every domain generated by two elements is a principal domain, is a PID

Published on 13 Nov 2018 - 12:34

1.4k

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Why $K[[X]]$ is PID and what's the form of the ring's ideals?

Published on 25 Mar 2026 - 18:57

249

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Position of $\mathbb{Z}[\sqrt{-d}]$ in class hierarchy for all integers $d$

Published on 25 Mar 2026 - 3:02

558

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Existence of Principal Ideal

Published on 16 Nov 2018 - 5:31

400

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When does $(a,b)=(\gcd(a,b))$ hold?

Published on 26 Mar 2026 - 8:14

180

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Is the product of two Smith Normal Forms a the Smith normal form of the product?

Published on 20 Mar 2026 - 1:01

140

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Is it possible for an integral domain to have an ideal that cannot be generated by a countable set?

Published on 25 Mar 2026 - 4:39

170

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Some counterexamples in basic ring theory

Published on 25 Mar 2026 - 7:38

560

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Examples of PIDs that are not Fields

Published on 12 Dec 2018 - 20:57

List Question

Principal ideal of a polynomial ring

PID controller for 2 DOF system

Let $D$ be a PID and $a$ and $b$ be nonzero elements of $D$. Prove that there exist elements $s$ and $t$ in $D$ such that $\gcd(a, b) = as + bt$.

non-PID which where all prime ideals are maximal

Proving Ascending Ideals Stabilize in PID

Prove that for distinct prime ideals $P, Q$ of PID $R$, $P^n + Q^m = R$.

Factorization Domain in which every domain generated by two elements is a principal domain, is a PID

Why $K[[X]]$ is PID and what's the form of the ring's ideals?

Position of $\mathbb{Z}[\sqrt{-d}]$ in class hierarchy for all integers $d$

Existence of Principal Ideal

When does $(a,b)=(\gcd(a,b))$ hold?

Is the product of two Smith Normal Forms a the Smith normal form of the product?

Is it possible for an integral domain to have an ideal that cannot be generated by a countable set?

Some counterexamples in basic ring theory

Examples of PIDs that are not Fields

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