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If in a UFD every maximal ideal is principal then it is a PID

score 2.9k · Answer 1 · 2026-03-29 07:39:23

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If in a UFD every maximal ideal is principal then it is a PID

Published on 29 Mar 2026 - 7:39

score 106 · Answer 2 · 2015-05-02 16:53:09

106

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$R$ commutative ring with unity. Prove if $R/M$ is a field $\implies M$ is a max ideal.

Published on 02 May 2015 - 16:53

score 514 · Answer 3 · 2015-05-03 02:31:01

514

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$R=\mathbb{Q}[x,y,z]$ and let bars denote passage to $\mathbb{Q}[x,y,z]/(xy-z^2)$. Prove that $\bar{P}=(\bar{x},\bar{z} )$ is a prime ideal.

Published on 03 May 2015 - 2:31

score 1.4k · Answer 4 · 2015-05-03 16:06:52

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A radical ideal in a commutative ring is prime if and only if it is not an intersection of two radical ideals properly containing it?

Published on 03 May 2015 - 16:06

score 628 · Answer 5 · 2015-05-05 06:57:23

628

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Number of distinct prime ideals

Published on 05 May 2015 - 6:57

score 203 · Answer 6 · 2015-05-08 10:21:41

203

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Why does $N(\mathfrak p)$ belong to $\mathfrak p$?

Published on 08 May 2015 - 10:21

score 498 · Answer 7 · 2015-05-09 14:54:16

498

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For a maximal left ideal $M$ of $S$, is $f^{-1}(M)$ a maximal left ideal of $R$ when $f$ is surjective?

Published on 09 May 2015 - 14:54

score 1.9k · Answer 8 · 2026-03-27 16:21:47

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how to tell if a ring is noetherian

Published on 27 Mar 2026 - 16:21

score 3k · Answer 9 · 2026-03-29 18:33:35

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bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$

Published on 29 Mar 2026 - 18:33

score 312 · Answer 10 · 2026-04-03 01:04:40

312

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Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Published on 03 Apr 2026 - 1:04

score 482 · Answer 11 · 2015-05-16 14:46:15

482

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Number of generators of prime ideals in $K[x_1,x_2,...,x_n]$

Published on 16 May 2015 - 14:46

score 189 · Answer 12 · 2026-03-30 01:51:35

189

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Residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$

Published on 30 Mar 2026 - 1:51

score 69 · Answer 13 · 2015-05-21 19:09:12

69

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Prime ideals in a quotient

Published on 21 May 2015 - 19:09

score 416 · Answer 14 · 2026-03-29 16:47:58

416

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Localising a polynomial ring and non-maximal prime ideal

Published on 29 Mar 2026 - 16:47

score 401 · Answer 15 · 2015-05-28 04:16:29

401

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Looking for a direct proof that all maximal ideals of $\mathbb C[x_1,x_2,...,x_n]$ are generated by $n$ linear polynomials

Published on 28 May 2015 - 4:16

List Question

If in a UFD every maximal ideal is principal then it is a PID

$R$ commutative ring with unity. Prove if $R/M$ is a field $\implies M$ is a max ideal.

$R=\mathbb{Q}[x,y,z]$ and let bars denote passage to $\mathbb{Q}[x,y,z]/(xy-z^2)$. Prove that $\bar{P}=(\bar{x},\bar{z} )$ is a prime ideal.

A radical ideal in a commutative ring is prime if and only if it is not an intersection of two radical ideals properly containing it?

Number of distinct prime ideals

Why does $N(\mathfrak p)$ belong to $\mathfrak p$?

For a maximal left ideal $M$ of $S$, is $f^{-1}(M)$ a maximal left ideal of $R$ when $f$ is surjective?

how to tell if a ring is noetherian

bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$

Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Number of generators of prime ideals in $K[x_1,x_2,...,x_n]$

Residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$

Prime ideals in a quotient

Localising a polynomial ring and non-maximal prime ideal

Looking for a direct proof that all maximal ideals of $\mathbb C[x_1,x_2,...,x_n]$ are generated by $n$ linear polynomials

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