List questions from principal-ideal-domains

Published on 09 Dec 2011 - 20:54

596

Views

How can we write this $\mathbb{Q}[x]$-module as a direct sum of cyclic $\mathbb{Q}[x]$-modules?

Published on 13 Dec 2011 - 2:42

345

Views

Number of ideals of a PID modulo an ideal

Published on 14 Dec 2011 - 1:33

4.4k

Views

Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

Published on 14 Dec 2011 - 21:48

7.8k

Views

Dedekind domain with a finite number of prime ideals is principal

Published on 27 Mar 2026 - 5:03

2.4k

Views

Are all subrings of the rationals Euclidean domains?

Published on 06 Jan 2012 - 14:44

396

Views

Queries on proof that every PID is a factorisation domain

Published on 08 Jan 2012 - 6:17

38.2k

Views

Show that every ideal of the ring $\mathbb Z$ is principal

Published on 22 Jan 2012 - 16:27

4.6k

Views

Greatest common divisor in the Gaussian Integers

Published on 22 Feb 2012 - 4:24

1.2k

Views

Characterization of primary ideals in a principal ideal domain

Published on 25 Feb 2012 - 22:14

181

Views

Why does the result hold for PIDs but not for UFDs?

Published on 09 Mar 2012 - 3:15

5.9k

Views

Show $\mathbb{Z}[\sqrt{2}]$ is a PID

Published on 12 Mar 2012 - 17:02

686

Views

Isomorphism of direct sums $R/a \oplus R/b \cong R/{\rm lcm}(a,b)\oplus R/\gcd(a,b)$

Published on 20 Mar 2012 - 7:42

622

Views

Rings such that $A[x]$ is a principal ideal domain

Published on 01 Apr 2012 - 13:12

7k

Views

A subring of the field of fractions of a PID is a PID as well.

Published on 28 Mar 2026 - 16:57

List Question

$R$ is PID, so $R/I$ is PID, and application on $\mathbb{Z}$ and $\mathbb{N}$

How can we write this $\mathbb{Q}[x]$-module as a direct sum of cyclic $\mathbb{Q}[x]$-modules?

Number of ideals of a PID modulo an ideal

Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

Dedekind domain with a finite number of prime ideals is principal

Are all subrings of the rationals Euclidean domains?

Queries on proof that every PID is a factorisation domain

Show that every ideal of the ring $\mathbb Z$ is principal

Greatest common divisor in the Gaussian Integers

Characterization of primary ideals in a principal ideal domain

Why does the result hold for PIDs but not for UFDs?

Show $\mathbb{Z}[\sqrt{2}]$ is a PID

Isomorphism of direct sums $R/a \oplus R/b \cong R/{\rm lcm}(a,b)\oplus R/\gcd(a,b)$

Rings such that $A[x]$ is a principal ideal domain

A subring of the field of fractions of a PID is a PID as well.

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