Question

$R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

score 104 · Answer 1 · 2016-07-24 04:55:35

104

Views

$R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

Published on 24 Jul 2016 - 4:55

score 461 · Answer 2 · 2026-03-25 11:04:55

461

Views

$R$ be an infinite commutative ring such that $R/I$ has only finitely many ideals for every non-zero ideal $I$ , what can we say about $R$?

Published on 25 Mar 2026 - 11:04

score 653 · Answer 3 · 2016-07-30 09:11:43

653

Views

Showing $1+3\sqrt{-5}$ is irreducible but not prime.

Published on 30 Jul 2016 - 9:11

score 2.4k · Answer 4 · 2016-08-02 08:01:43

2.4k

Views

Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime

Published on 02 Aug 2016 - 8:01

score 701 · Answer 5 · 2016-08-09 12:18:03

701

Views

Does every ideal of the ring of all algebraic integers contain a finite product of prime ideals?

Published on 09 Aug 2016 - 12:18

score 1.9k · Answer 6 · 2016-08-14 22:36:09

1.9k

Views

If $A$ and $B$ are integral domains, how to make $A\times B$ an integral domain?

Published on 14 Aug 2016 - 22:36

score 253 · Answer 7 · 2016-08-21 06:41:20

253

Views

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

Published on 21 Aug 2016 - 6:41

score 27 · Answer 8 · 2016-08-21 16:17:10

27

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 21 Aug 2016 - 16:17

score 27 · Answer 9 · 2026-03-25 06:05:43

27

Views

Characteristic of a non unital integral domain

Published on 25 Mar 2026 - 6:05

score 201 · Answer 10 · 2016-08-23 14:56:41

201

Views

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

Published on 23 Aug 2016 - 14:56

score 1.5k · Answer 11 · 2016-08-27 16:43:10

1.5k

Views

Are rank 1 projective modules over domains isomorphic to ideals of R?

Published on 27 Aug 2016 - 16:43

score 294 · Answer 12 · 2016-08-28 15:10:27

294

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 28 Aug 2016 - 15:10

score 162 · Answer 13 · 2016-08-29 11:59:34

162

Views

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

Published on 29 Aug 2016 - 11:59

score 217 · Answer 14 · 2016-08-29 14:55:21

217

Views

Let $R$ be an integral domain which is not a field. Then can $R[x]$ have a maximal ideal generated by a non-constant polynomial?

Published on 29 Aug 2016 - 14:55

score 42 · Answer 15 · 2016-09-02 09:41:41

42

Views

Equivalent conditions on $d$ being a common divisor of $a$ and $b$ in an integral domain: $(a,b) \subseteq (d) \Leftrightarrow d|a$ and $d|b$

Published on 02 Sep 2016 - 9:41

List Question

$R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

$R$ be an infinite commutative ring such that $R/I$ has only finitely many ideals for every non-zero ideal $I$ , what can we say about $R$?

Showing $1+3\sqrt{-5}$ is irreducible but not prime.

Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime

Does every ideal of the ring of all algebraic integers contain a finite product of prime ideals?

If $A$ and $B$ are integral domains, how to make $A\times B$ an integral domain?

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?

Characteristic of a non unital integral domain

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

Are rank 1 projective modules over domains isomorphic to ideals of R?

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

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

Let $R$ be an integral domain which is not a field. Then can $R[x]$ have a maximal ideal generated by a non-constant polynomial?

Equivalent conditions on $d$ being a common divisor of $a$ and $b$ in an integral domain: $(a,b) \subseteq (d) \Leftrightarrow d|a$ and $d|b$

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