Questions tagged [principal-ideal-domains]
For questions about principal ideal domains: rings without zero divisors where every ideal is principal.
735
questions
0
votes
0answers
28 views
understanding the contradiction of $I$ is not free.
My professor gave us this example on a module that is not free :
$R = k[x,y,z]$ where $k$ is a field. $I = xyR + yz R + xz R \subset R.$ take $u = xy, v = yz, w = xz$ then $I = uR + vR + wR.$ And so, ...
0
votes
1answer
26 views
Ideals of quotient ring of a principal ideal domain are also principal ideals.
Let $R$ be a PID. I have to show that every ideal of a quotient ring of $R$ is a principal ideal. I am able to visualize this problem by taking $R=Z$. I took ideal $(20)$ of $Z$ for example. Then I ...
3
votes
1answer
51 views
Relations between Cayley-Hamilton theorem and the classification theorem for finitely generated modules over a PID
There is a sentence in the book "Algebra Chapter 0" that the Cayley-Hamilton theorem would be evident if we connect it with the classification theorem for finitely generated modules over a ...
8
votes
1answer
90 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
There are number of similar questions to this, but I have read through them all and the answers either rely on results that I don't have access to or I'm unsure how to translate them to this situation....
1
vote
0answers
33 views
What kind of subrings inherit their parent ring's properties?
I was studying the proof my teacher from my abstract algebra course gave me for the Gauss Lemma, and I came up with a question he could not answer me at that moment (it's the first time he works ...
0
votes
0answers
42 views
irrationality of $\sqrt{2}$ with ring theory [duplicate]
I consider the ideal $I \subseteq \Bbb Z$ with $I=\{n \in \Bbb Z \mid n \times \sqrt{2} \in \Bbb Z\}$. I do not manage to prove that this ideal is trivial. Can you help me ? Thanks.
2
votes
2answers
31 views
Find a generator of the ideal generated by $3$ and $2-2\sqrt{-2}$ in ${\mathbb{Z}\left[\sqrt{-2}\right]}$
Exercise: Find a generator of the ideal generated by $3$ and $2-2\sqrt{-2}$ in ${\mathbb{Z}\left[\sqrt{-2}\right]}$.
I know that such a generator exists since $\mathbb{Z}\left[\sqrt{-2}\right]$ is a ...
1
vote
1answer
45 views
maximal ideal of PIR
PIR means principal ideal ring.This is extended concept of PID, we do not assume integral domain.
Is it true that every nonzero prime ideal is maximal, even in PIR ?
Or, do we need to restrict the ...
0
votes
1answer
28 views
For commutative rings, if the ring of polynomials is a PID then the ring is an integral domain.
I am reading Dummit and Foote's proof that if $R$ is a commutative ring, and if $R[x]$ is a PID, then $R$ is a field. At the start of the proof they write
Proof: Assume $R[x]$ is a Principal Ideal ...
1
vote
2answers
63 views
Can you suggest some challenging ring theory problems? [closed]
I would like to work on some wonderful excercises (not particularly hard) in ring theory, subjects PID, UFD or polynomial rings.
Can you introduce a resource that has newer exercises?
Or if you can ...
2
votes
1answer
28 views
Show that the ideal $AB$ is prime iff $A = \{0_R\}$ or $B = \{0_R\}$ as follows [closed]
Let $A$ and $B$ be ideals in a principal ideal domain $R$, where $A \ne R$ and $B \ne R$.
Show that the ideal $AB$ is prime iff $A = \{0_R\}$ or $B = \{0_R\}$.
I know that since $R$ is PID then $A$ ...
1
vote
2answers
69 views
Is there a field $K$ for which the kernel of the map $\mathbb{Z}[x]\to K$ is NOT principal?
Consider $K$ a field and a ring homomorphism $\phi:\mathbb{Z}[x]\to K$.
I guess this is wrong, because I was asked to give an example of precisely a field $K$ for which the kernel of the map above is ...
0
votes
1answer
29 views
Is PID cyclic? I cannot understand quotient ring is cyclic
In the textbook, it says
āIf $R$ is a principal ideal domain and $r$ is an elt of $R$, then the quotient ring $R/(r)$ is a cyclic R-module.
I dont know why $R/(r)$ is cyclic.
Thank you in advance.
1
vote
1answer
53 views
Prove that ring of fractions of $Q[x]$ is a PID
I'm trying to prove the following:
Let
$$
R = \left\{\frac{f(x)}{g(x)}\mid f(x),g(x)\in\mathbb{Q}[x], g(x)\not=0\right\}\subset\mathbb{Q}(x)
$$
Show that $R$ is a principal ideal domain.
My approach ...
1
vote
0answers
45 views
Generalization of Bezout Identity for Polynomials
Let $i \in \{1,\ldots, n\}$, $f_i(x)$ be a univariate polynomial, and $g(x) = \mathsf{GCD}(f_1(x), \ldots,f_n(x))$. According to Bezout identity, there exists $a_i(x)$ such that:
$$\sum_{i \in [n]}a_i(...
0
votes
0answers
28 views
Order of a Module
Here is the problem I am working on:
If $M$ is a finitely generated torsion module over a PID $R$ such that there exists $m \in M$ with $ann(m) = (r)$, where $R \ni r = order(M)$, what can you say ...
0
votes
1answer
49 views
Is $(1+x^2,x^3) \in \mathbb{Q}[x]$ a principal ideal?
Is this line of reasoning correct?
If $\big(1+x^2,\:x^3\big) \in \mathbb{Q}[x]$ were a principal ideal, then there would be a polynomial $f\in \mathbb{Q}[x]$ such that $\big(f(x)\big) = \big(1+x^2,x^3\...
0
votes
1answer
34 views
Why irreduciblilty and not division leads to relative primeness in a PID
I'm unable to understand reasoning behind following conclusion: If $p$ is irreducible element of PID $R$ and $p$ doesn't divide $a$ of R. Then $p$ and $a$ are relatively prime.
The definition of a ...
0
votes
1answer
30 views
Ideal in principal ideal domain
Let $R$ be a principal ideal domain and let $a$ and $b$ be two nonunit elements in $R$ then ideal generated by $a$ and $b$ is also generated by
$a+b$
$ab$
$\gcd(a,b)$
$\text{lcm}(a,b)$
I am not ...
2
votes
1answer
82 views
Units in a ring between an integral domain $R$ and its fraction field.
Let $R$ be an integral domain with fraction field $K$, and let $\mathcal{T}$ be the set of subrings $T$ of $K$ such that $R\subseteq T$.
Call $R\;$unit-complete$\;$if for all $T\in \mathcal{T}$, ...
1
vote
0answers
19 views
Prove that $x = uy$ for some unit $u \in R^{\times}$ iff $(x)=(y)$ as ideals. [duplicate]
Prove that:In a commutative ring $R,$ $x = uy$ for some unit $u \in R^{\times}$ iff $(x)=(y)$ as ideals.
My thoughts:
I feel like the idea is similar to the proof of the following proposition:
...
2
votes
1answer
115 views
Let $F$ be an infinite field and let $f(x) ā F[x]$. If $f(a) = 0$ for infinitely many $a ā F$, show that $f = 0$. [duplicate]
Step 1:
Suppose that $F$ is an infinite field and $f(x) \in F[x]$.
To claim the statement,
"If $f(a)=0$ for infinitely many elements $a$ of $F$, then $f(x)=0$".
To prove this statement using ...
1
vote
1answer
27 views
Reference on different types of integral domains
I am looking for a good reference (book or otherwise) which has a comprehensive study of the different types of integral domains, including:
Euclidean domains
UFDs
GCD domains
PIDs
Dedekind domains
...
0
votes
0answers
27 views
The difference in the procedure of showing that every ideal is principal in $\mathbb{Z}$ and in $\mathbb{Z}[\sqrt{3}]$?
I am trying to prove that every ideal is principal in $\mathbb{Z}[\sqrt{3}].$ And I got a hint to use the norm function and the following division property satisfied by $\mathbb{Z}[\sqrt{3}]$:
Given $...
1
vote
0answers
52 views
Uniqueness of a Decomposition of a finitely generated torsion module over a PID
Let $M$ be a finitely generated torsion $R$-module. Show that we can write $M \cong R/(a_1) \oplus ... \oplus R/(a_n)$ such that $a_1 | a_2 | ... | a_n$. Show that $a_1,...,a_n$ are uniquely ...
1
vote
1answer
33 views
How to prove $ y_1, \dots , y_n$ generate module Mļ¼
Let $M$ be a finitely generated module over PID $R$, and $M$ is generated by $x_1, x_2, \dots, x_n$. Let $y_1=a_1\cdot x_1+\dots +a_n\cdot x_n$, and $\gcd(a_1, \dots , a_n)=1.$ Prove that there exist $...
1
vote
0answers
69 views
Isomorphism between posets of ideals of $\mathbb Q[x]$ and $\mathbb Q[x,y]$.
Here is a problem that I have only solved partially. I am (edit: "was") stuck at the second part.
Problem: Given that two posets $(\mathcal{P}_1,\leq_1)$ and $(\mathcal{P}_2,\leq_2)$ are ...
1
vote
0answers
49 views
What condition makes $\mathbb{Z}[a]$ is PID, where $a$ is complex number.
I want to find the condition which $a$ makes $\mathbb{Z}[a]$ is a PID(Principal Ideal Domain). I wonder am I right.
My answer is :
If $R$ is Noetherian ring, then $R[x]$ is also Noetherian ring. As ...
0
votes
2answers
75 views
Which element belong to the principal ideal $(3+i)$?
In the ring $\mathbb{Z}[i]\space$, which of the following elements belongs to the principal ideal $(3+i)$?
(a) $1+8i$
(b) $1+5i$
(c) $1+6i$
(d) $1+7i$
1
vote
1answer
37 views
Proof that Divisible Modules are Injective over a PID
I'm reading the proof given in Hilton/Stammbach's homological algebra book that over a PID, injective modules and divisible modules are the same. (Thm. 7.1 in chapter 1, pp. 31-32). I'm stuck on a ...
1
vote
2answers
99 views
Maximal free submodule over a PID
Let M be a f.g. module over a p.i.d. and T(M) be its torsion submodule. Then M is the direct sum of T(M) and of a free submodule F, unique up to isomorphism, and in addition which is a maximal free ...
3
votes
2answers
75 views
Are quotients of a PID by non-prime ideals ever a PID? [duplicate]
I've proved that any quotient of a PID by a prime ideal is again a PID as an exercise, and got into thinking about quotients by non-prime ideals. Are they ever a PID? More concretely, say $R$ is a PID ...
1
vote
0answers
52 views
What is the dimension of this vector space?
This theorem is to show the uniqueness of the Structure Theorem
Can someone elaborate the part where they deduce the $\dim M(p)?$
For context let me elaborate the previous sentence preceding the ...
0
votes
1answer
44 views
Why does the ring element get absorbed here?
Suppose $M$ is finitely generated over $R$ that is a PID. Let $x \in M$, $\ann(x) = (a)$, $p\in R$ be irreducible. Then
If $p|a$ then $Rx/pRx \approx R/(p)$
If $p \not| a$, then $pRx = Rx.$
The ...
1
vote
0answers
168 views
How to show that ${\Bbb Q}[x,y]/(x^2+y^2+1)$ is a PID?
I know that ${\Bbb Q}[x,y]/(x^2+y^2-1)$ is not a PID, because it is not a UFD.
But ${\Bbb Q}[x,y]/(x^2+y^2+1)$ is a PID.
I heard about some proof using Hochschild-Serre spectral sequence. I was ...
1
vote
1answer
25 views
Decomposition of finitely generated Module over PID
Suppose
Let $M = <x,y>/<2x-3y> $ where $<x,y>, <2x-3y>$ are $ \mathbb{Z} $-modules.
How can I find the decomposition of $M$ to its invariant factors?
1
vote
1answer
42 views
Computing $\operatorname{Aut}(\mathbb{C}[x]/(x-a)^n)$
How do I find $\operatorname{Aut}(\mathbb{C}[x]/(x-a)^n)$ as $\mathbb{C}[x]$ module? Does it have a reasonable description?
3
votes
1answer
77 views
Exact sequences and characteristic ideal of modules over a principal ideal domain
Let $R$ be a principal ideal ring, which is not a field, and $M$ be a torsion module, then there is an isomorphism due to the theorem for finitely generated modules over a principal domain
$ M \cong \...
0
votes
2answers
134 views
Let $M$ be a free module over a PID with finite rank, then any submodule $N \subset M$ is also free with finite rank
I want to post the statement from Dummit and Foote (page460), but it is so long that I have to post a shorter one from another book.
Q2: I want to make sure I understand this. We apply induction on $...
1
vote
2answers
57 views
A question about PIDs
Every PID $R$ is Noetherian, since if $a_1R\subsetneq a_2R\subsetneq \cdots$ is an infinite increasing chain of ideals, then the union $I=\cup_i a_iR$ is also an ideal, hence $I=bR$ for some $b\in R$. ...
0
votes
0answers
63 views
In $\mathbb Z[x]$, show that the set $A$ of all polynomials with even constant term is not a principal ideal
I'm doing Exercise 1 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
In $\mathbb Z[x]$, show that the set $A$ of all polynomials with even constant term is not a principal ideal.
...
0
votes
1answer
36 views
Polynomial Ring of a Field extension
The problem is stated as follows: Let $K$ be a finite field extension of $F$ and consider $P\in K[x]$ monic irreducible. Then, there exists $Q\in K[x]$ such that $PQ\in F[x]$.
This question is ...
2
votes
1answer
240 views
$(1-x,y)$ is not principal in $\Bbb Q[x,y]/(x^2+y^2-1)$
I'm very interested in this interesting approach to the problem whether $\Bbb Q[x,y]/(x^2+y^2-1)$ is PID or not.
https://math.stackexchange.com/questions/3573915/1-x-y-is-not-principal-in-bbb-qx-y-x2ļ¼...
2
votes
0answers
55 views
Ad hoc proof of Minkowski's bound for $\mathbb{Q}(\sqrt{-19})$
Let $R=\mathbb{Z}[\frac{-1+\sqrt{-19}}{2}]=\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt{-19})$.
A classical way to prove the fact that $R$ is a PID is to establish that for all $a,b\in R,b\neq 0$, there ...
1
vote
1answer
132 views
Locally Principal Ideal Domain [duplicate]
Let $A$ be a domain with $\max(A)$ a finite set. It is known that if $\mathfrak{m}\in \max(A)$ then $A_{\mathfrak{m}}$ is a PID.
Show that $A$ is also a PID.
Here is what I was thinking. Suppose that ...
1
vote
0answers
55 views
Show that every ideal in the ring of Gaussian integers is principal
I'm doing Exercise 9 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
Show that every ideal in the ring of Gaussian integers is principal.
Could you please verify if my attempt is fine ...
2
votes
1answer
136 views
Proof of every PID is Noetherian
I saw the proof of this proposition in here, but I have a question about this.
Definition of Noetherian ring is that ring is commutative, and every ideal of R is finitely generated, right? Principal ...
1
vote
1answer
57 views
What is principal ideal generated by $p$?
I'm studying abstract algebra, with Dummit's book.
Our professor introduced a lemma, and I'm confused with some concept.
Here is the lemma: "Let $R$ be an integral domain, and let $p$ be in $R$. ...
8
votes
1answer
128 views
uncountable principal ideal domain with few units
Does there exist an uncountable principal ideal domain with only countably many units (or, even, with only finitely many units) ?
The answer would be yes, if only unique factorization domain was ...
1
vote
1answer
68 views
Question about Principal ring
I'm mathematics student. I just confusing about some basic concept, so I ask for it here...
First of all, principal ideal means ideal generated by a single element, but I don't know what 'generated' ...
