Question

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 $I=(0)$?

score 847 · Answer 1 · 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 844 · Answer 2 · 2016-08-27 11:59:04

844

Views

Show $\mathbb{Z}G$ is a noetherian ring for finite group $G$

Published on 27 Aug 2016 - 11:59

score 65 · Answer 3 · 2016-08-27 21:20:59

65

Views

Does an ideal in $k[x_1, \dots, x_n]$, $k$ a field, contain only finitely many irreducible elements?

Published on 27 Aug 2016 - 21:20

score 138 · Answer 4 · 2026-03-25 11:04:55

138

Views

$C$ be a subring of $B$ which is again a subring of $A$ , let $A,B,C$ be Noetherian and $A \cong C$ , then is $A \cong B$?

Published on 25 Mar 2026 - 11:04

score 163 · Answer 5 · 2026-03-26 11:05:06

163

Views

Does there exist a Noetherian domain (which is not a field) whose field of fractions is (isomorphic with) $\mathbb C$?

Published on 26 Mar 2026 - 11:05

score 41 · Answer 6 · 2016-08-31 16:29:07

41

Views

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

Published on 31 Aug 2016 - 16:29

score 412 · Answer 7 · 2016-09-07 08:53:40

412

Views

If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$

Published on 07 Sep 2016 - 8:53

score 155 · Answer 8 · 2016-09-07 15:42:40

155

Views

$I$ be an ideal of a Noetherian ring $R$ , then does there exists $ n \in \mathbb N$ such that $(\sqrt I)^n \subseteq I$ ?

Published on 07 Sep 2016 - 15:42

score 1.1k · Answer 9 · 2016-09-09 08:00:34

1.1k

Views

Is the function ring $C^{\infty}(M)$ noetherian?

Published on 09 Sep 2016 - 8:00

score 339 · Answer 10 · 2016-09-10 12:56:38

339

Views

Proving that $A$ is a Noetherian ring when $M$ is a faithful Noetherian $A$-module

Published on 10 Sep 2016 - 12:56

score 764 · Answer 11 · 2016-09-11 13:37:00

764

Views

Unique minimal prime over an ideal

Published on 11 Sep 2016 - 13:37

score 424 · Answer 12 · 2016-09-21 14:17:22

424

Views

Power of an ideal in a Noetherian ring

Published on 21 Sep 2016 - 14:17

score 978 · Answer 13 · 2016-09-22 06:22:44

978

Views

A reduced Noetherian ring injects into a product of fields

Published on 22 Sep 2016 - 6:22

score 365 · Answer 14 · 2026-03-25 06:11:15

365

Views

An ascending chain of subgroups

Published on 25 Mar 2026 - 6:11

score 636 · Answer 15 · 2026-03-25 11:00:08

636

Views

About Fitting decomposition theorem of module

Published on 25 Mar 2026 - 11:00

List Question

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 $I=(0)$?

Show $\mathbb{Z}G$ is a noetherian ring for finite group $G$

Does an ideal in $k[x_1, \dots, x_n]$, $k$ a field, contain only finitely many irreducible elements?

$C$ be a subring of $B$ which is again a subring of $A$ , let $A,B,C$ be Noetherian and $A \cong C$ , then is $A \cong B$?

Does there exist a Noetherian domain (which is not a field) whose field of fractions is (isomorphic with) $\mathbb C$?

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

If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$

$I$ be an ideal of a Noetherian ring $R$ , then does there exists $ n \in \mathbb N$ such that $(\sqrt I)^n \subseteq I$ ?

Is the function ring $C^{\infty}(M)$ noetherian?

Proving that $A$ is a Noetherian ring when $M$ is a faithful Noetherian $A$-module

Unique minimal prime over an ideal

Power of an ideal in a Noetherian ring

A reduced Noetherian ring injects into a product of fields

An ascending chain of subgroups

About Fitting decomposition theorem of module

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