Question

A UFD such that every prime ideal is contained in a principal proper ideal is a PID?

score 254 · Answer 1 · 2026-03-26 14:22:49

254

Views

A UFD such that every prime ideal is contained in a principal proper ideal is a PID?

Published on 26 Mar 2026 - 14:22

score 28 · Answer 2 · 2026-03-26 11:01:40

28

Views

Domain $R$ s.t. for any proper ideal $I$ , $\mathcal F_I:=\{(x):x\in R , I \subseteq (x) \ne R\}$ is non-empty implies it contains a minimal element?

Published on 26 Mar 2026 - 11:01

score 202 · Answer 3 · 2026-03-26 11:00:23

202

Views

Can we characterize all infinite PID s whose group of units is singleton?

Published on 26 Mar 2026 - 11:00

score 269 · Answer 4 · 2016-08-23 17:11:42

269

Views

For any set in a principal ideal ring, can one choose a finite subset so that the generated ideals are the same?

Published on 23 Aug 2016 - 17:11

score 391 · Answer 5 · 2026-03-31 14:30:46

391

Views

Number of generators of an ideal of the polynomial ring over a field

Published on 31 Mar 2026 - 14:30

score 847 · Answer 6 · 2026-03-26 01:23:39

847

Views

Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?

Published on 26 Mar 2026 - 1:23

score 100 · Answer 7 · 2016-08-28 01:18:11

100

Views

Calculating the radical of a simple ideal

Published on 28 Aug 2016 - 1:18

score 495 · Answer 8 · 2026-03-26 01:12:16

495

Views

$R$ is a subring of $A$ which is a subring of $R[x,y]$ then does there exist an ideal $J$ of $A$ such that $A/J$ and $R$ are ring isomorphic?

Published on 26 Mar 2026 - 1:12

score 295 · Answer 9 · 2026-03-26 11:00:01

295

Views

$D$ be a UFD having infinitely many maximal ideals , then does $D$ have infinitely many irreducible elements which are pairwise non-associate?

Published on 26 Mar 2026 - 11:00

score 128 · Answer 10 · 2016-08-28 23:24:19

128

Views

Prove that $\mathbb{Z \times Z}/ \{(3m,n)\in\mathbb{Z \times Z}:m,n \in \mathbb{Z}\} $ is a field.

Published on 28 Aug 2016 - 23:24

score 234 · Answer 11 · 2016-08-29 13:41:39

234

Views

Contracting principal ideals in univariate polynomial rings to subrings generated by a monomial

Published on 29 Aug 2016 - 13:41

score 129 · Answer 12 · 2026-03-31 12:16:35

129

Views

Radical of an ideal in the ring $\mathbb{Z^{\mathbb{N}}}$

Published on 31 Mar 2026 - 12:16

score 876 · Answer 13 · 2016-08-29 19:22:05

876

Views

Ideal generated by two coprime polynomials in $\mathbb{Z}[x]$

Published on 29 Aug 2016 - 19:22

score 82 · Answer 14 · 2026-03-29 21:34:19

82

Views

An equality between right ideals and Jacobson radical

Published on 29 Mar 2026 - 21:34

score 42 · Answer 15 · 2026-03-26 16:05:47

42

Views

Is the module $\left<1,x,\dots, x^{s-1}\right>$ noetherian?

Published on 26 Mar 2026 - 16:05

List Question

A UFD such that every prime ideal is contained in a principal proper ideal is a PID?

Domain $R$ s.t. for any proper ideal $I$ , $\mathcal F_I:=\{(x):x\in R , I \subseteq (x) \ne R\}$ is non-empty implies it contains a minimal element?

Can we characterize all infinite PID s whose group of units is singleton?

For any set in a principal ideal ring, can one choose a finite subset so that the generated ideals are the same?

Number of generators of an ideal of the polynomial ring over a field

Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?

Calculating the radical of a simple ideal

$R$ is a subring of $A$ which is a subring of $R[x,y]$ then does there exist an ideal $J$ of $A$ such that $A/J$ and $R$ are ring isomorphic?

$D$ be a UFD having infinitely many maximal ideals , then does $D$ have infinitely many irreducible elements which are pairwise non-associate?

Prove that $\mathbb{Z \times Z}/ \{(3m,n)\in\mathbb{Z \times Z}:m,n \in \mathbb{Z}\} $ is a field.

Contracting principal ideals in univariate polynomial rings to subrings generated by a monomial

Radical of an ideal in the ring $\mathbb{Z^{\mathbb{N}}}$

Ideal generated by two coprime polynomials in $\mathbb{Z}[x]$

An equality between right ideals and Jacobson radical

Is the module $\left<1,x,\dots, x^{s-1}\right>$ noetherian?

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