Question

Is $(1+x^2,x^3) \in \mathbb{Q}[x]$ a principal ideal?

score 61 · Answer 1 · 2020-11-29 20:31:29

61

Views

Is $(1+x^2,x^3) \in \mathbb{Q}[x]$ a principal ideal?

Published on 29 Nov 2020 - 20:31

score 110 · Answer 2 · 2026-03-29 09:10:10

110

Views

Generalization of Bezout Identity for Polynomials

Published on 29 Mar 2026 - 9:10

score 327 · Answer 3 · 2020-12-09 16:37:05

327

Views

Prove that a ring of fractions of $\mathbb Q[x]$ is a PID

Published on 09 Dec 2020 - 16:37

score 300 · Answer 4 · 2020-12-12 08:31:29

300

Views

Is PID cyclic? I cannot understand quotient ring is cyclic

Published on 12 Dec 2020 - 8:31

score 191 · Answer 5 · 2020-12-14 19:00:44

191

Views

Is there a field $K$ for which the kernel of the map $\mathbb{Z}[x]\to K$ is NOT principal?

Published on 14 Dec 2020 - 19:00

score 96 · Answer 6 · 2020-12-17 03:02:51

96

Views

Show that the ideal $AB$ is prime iff $A = \{0_R\}$ or $B = \{0_R\}$ as follows

Published on 17 Dec 2020 - 3:02

score 185 · Answer 7 · 2026-03-30 10:36:54

185

Views

Can you suggest some challenging ring theory problems?

Published on 30 Mar 2026 - 10:36

score 352 · Answer 8 · 2026-03-29 08:15:09

352

Views

For commutative rings, if the ring of polynomials is a PID then the ring is an integral domain.

Published on 29 Mar 2026 - 8:15

score 145 · Answer 9 · 2021-01-11 12:05:50

145

Views

maximal ideal of PIR

Published on 11 Jan 2021 - 12:05

score 92 · Answer 10 · 2026-03-27 05:59:35

92

Views

Find a generator of the ideal generated by $3$ and $2-2\sqrt{-2}$ in ${\mathbb{Z}\left[\sqrt{-2}\right]}$

Published on 27 Mar 2026 - 5:59

score 43 · Answer 11 · 2021-01-17 18:34:56

43

Views

irrationality of $\sqrt{2}$ with ring theory

Published on 17 Jan 2021 - 18:34

score 117 · Answer 12 · 2026-03-29 06:33:47

117

Views

What kind of subrings inherit their parent ring's properties?

Published on 29 Mar 2026 - 6:33

score 276 · Answer 13 · 2021-01-28 23:52:59

276

Views

$A \oplus B \cong A \oplus C$ implies $B \cong C$ where $A,B,C$ are finitely generated $R$ modules and $R$ is a PID

Published on 28 Jan 2021 - 23:52

score 148 · Answer 14 · 2021-02-01 22:32:20

148

Views

Relations between Cayley-Hamilton theorem and the classification theorem for finitely generated modules over a PID

Published on 01 Feb 2021 - 22:32

score 825 · Answer 15 · 2021-02-20 10:16:57

825

Views

Ideals of quotient ring of a principal ideal domain are also principal ideals.

Published on 20 Feb 2021 - 10:16

List Question

Is $(1+x^2,x^3) \in \mathbb{Q}[x]$ a principal ideal?

Generalization of Bezout Identity for Polynomials

Prove that a ring of fractions of $\mathbb Q[x]$ is a PID

Is PID cyclic? I cannot understand quotient ring is cyclic

Is there a field $K$ for which the kernel of the map $\mathbb{Z}[x]\to K$ is NOT principal?

Show that the ideal $AB$ is prime iff $A = \{0_R\}$ or $B = \{0_R\}$ as follows

Can you suggest some challenging ring theory problems?

For commutative rings, if the ring of polynomials is a PID then the ring is an integral domain.

maximal ideal of PIR

Find a generator of the ideal generated by $3$ and $2-2\sqrt{-2}$ in ${\mathbb{Z}\left[\sqrt{-2}\right]}$

irrationality of $\sqrt{2}$ with ring theory

What kind of subrings inherit their parent ring's properties?

$A \oplus B \cong A \oplus C$ implies $B \cong C$ where $A,B,C$ are finitely generated $R$ modules and $R$ is a PID

Relations between Cayley-Hamilton theorem and the classification theorem for finitely generated modules over a PID

Ideals of quotient ring of a principal ideal domain are also principal ideals.

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