For questions about principle ideal domains: rings without zero divisors where every ideal is principle.
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2answers
48 views
radical of sum of two ideals
$I$ and $J$ are ideals in $k[x_1,\cdots,x_n]$.
Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$.
I have no idea how to prove it. Can someone help?
0
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2answers
62 views
Show that if $D$ is a domain but not a field then $D[x]$ is not a principal ideal domain. [duplicate]
Show that if $D$ is a domain but not a field then $D[x]$ is not a principal ideal domain.
Sorry for my english....
4
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3answers
81 views
Show that $\mathbb{Z}[x]=\lbrace \sum_{i=0}^{n}{a_ix^i}:a_i \in \mathbb{Z}, n \geq 0 \rbrace$ is not a principal ideal ring.
Show that $\mathbb{Z}[x]=\lbrace \sum_{i=0}^{n}{a_ix^i}:a_i \in \mathbb{Z}, n \geq 0 \rbrace$ is not a principal ideal ring. I know the definition of principal ideal ring is that every ideal is ...
2
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1answer
74 views
which of the following statements are true and why?
which of the following statements are true and why?
Any two irreducibles in any UFD are associates.
If $D$ is a PID, then $D[x]$ is a PID.
In any UFD, if $p|a$ for an irreducible $p$, then $p$ ...
1
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1answer
90 views
Domain of a complex function
1) Why do the domain of a complex function has to be a disk (circular neighborhood of Zo)?
$$
|z-z_0|<p
$$
2) Domain is an open connected set. An open set D is said to be connected if every pair of ...
0
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2answers
152 views
Show that $\mathbb{Z}[\theta]$ (where $\theta = (1 + \sqrt{19}i)/2$) is a principal ideal domain.
I'm having difficulties with a homework problem from Algebra by Hungerford.
Let $R$ be the following subring of the complex numbers: $R = \{a + b(1 + \sqrt{19}i)/2 \mid a, b \in \mathbb{Z}\}$. ...
3
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1answer
115 views
Units in $\mathbb{Z}[\sqrt[3]{2}]$ : $\pm(1+\sqrt[3]{2}+(\sqrt[3]{2})^2)^n$?
The subring $\mathbb{Z}[\sqrt[3]{2}]\subset\mathbb{C}$ is a PID. I remember reading somewhere that the units in $\mathbb{Z}[\sqrt[3]{2}]$ are precisely the elements ...
5
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1answer
147 views
Quotient of ring of integers
Let $R=\mathcal{O}(K)$ be the ring of the integers of $K=\mathbb{Q}[\zeta_8]$, where $\zeta_8=e^{2\pi i/8}=\sqrt{2}/2(1+i)$ is a primitive eighth root of unity in $\mathbb{C}$. It can be shown that ...
0
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1answer
46 views
“Two ways” to reduce a module
Let $M$ be a module over a principal ideal domain $R$ and $\mathfrak{m}$ a maximal ideal of $R$ with residue field $R/\mathfrak{m}=k$ of characteristic $p$.
Under what circumstances are the modules
...
9
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5answers
329 views
How does a Class group measure the failure of Unique factorization?
I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
2
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1answer
65 views
Reference on Operators on Modules over PID
The operators on finite dimensional vector spaces over any field can be seen in nice form w.r.t. some choice of basis such as "diagonal, triangular, Jordan form, rational form etc." But, for ...
4
votes
1answer
86 views
Is $(x^3-x^2+2x-1)$ prime in $\mathbb{Z}/(3)[x]$?
This is somewhat of a follow up on this question: Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}[x]$?
I'm curious, is $\mathbb{Z}[x]/I$ a domain, with $I=(3,x^3-x^2+2x-1)$? I know $I$ is not ...
2
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0answers
190 views
Exercise on modules over PID involving injective modules, Baer's criterion.
I'm interested if I solved this somewhat correctly, and would like to be set straight if it is wrong. This is an exercise from an introductory text.
Let $A$ be a module over a principal ideal ...
2
votes
2answers
460 views
Every prime ideal is either zero or maximal in a PID.
$(1)$ Let $R$ be a commutative ring with $1\neq 0.$ If $R$ is a PID, show that every prime ideal is either zero or maximal.
In many books I have found the proof of the above statement where they ...
3
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4answers
291 views
When an integral domain is a PID?
Does anyone has a simple proof of the following fact: "An integral domain whose every (nonzero?) prime ideal is principal is a principal ideal domain"?
6
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1answer
139 views
Integer extensions, rings $\mathbb{Z}[\sqrt{s}]$
I'm not sure if this type of question is acceptable here, but I'd really appreciate someone's help. I'm about to start writing a semestral work that we need to achieve the Bachelor's degree in our ...
3
votes
2answers
164 views
A question about Euclidean Domain
This is a problem from Aluffi's book, chapter V 2.17.
"Let $R$ be a Euclidean Domain that is not a field. Prove that there exists a nonzero, nonunit element $c$ in $R$ such that $\forall a \in R$, ...
2
votes
1answer
129 views
Ring of analytic functions on the circle
Let $A = C^\omega(S^1)$ (resp. $C^\omega_{\mathbb C}(S^1)$) the ring of real-analytic real-valued (resp. complex valued) functions on the circle.
These rings have maximal ideals $\mathfrak m_p = ...
2
votes
4answers
134 views
If two elements in a ED have the same Euclidean norm, they are associates?
Is it very obvious that on a Euclidean Domain, two elements $x$ and $y$ have the same Euclidean norm $\nu(x) = \nu(y)$ then they are associates?
Can someone give me a proof of this?
0
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3answers
651 views
Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain.
Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain.
We haven't really discussed principal ideal domains. I've heard that this is easy, but I just ...
0
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1answer
91 views
abstract algebra question concerning Euclidean domains
Let $R$ be a Euclidean domain, and let $r_{1}$, $r_{2}$, $r_{3}, \ldots,r_{n}$ be (distinct) elements of $R$. Prove that there are elements $a_{1}$, $a_{2}$, $a_{3},\ldots,a_{n}$ such that $d = ...
1
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1answer
181 views
Rings such that $A[x]$ is a principal ideal domain
Let $A$ be a commutative ring.
Then the following assertions are equivalent.
$A$ is a field;
$A[x]$ is a Euclidean domain;
$A[x]$ is a principal ideal domain;
$A[x]$ is a unique factorization ...
1
vote
1answer
677 views
Show $\mathbb{Z}[\sqrt{2}]$ is a PID
I understand how to show something isn't a PID (namely by constructing a counterexample), and I think I understand the proof that $\mathbb{Z}$ is a PID, but I'm not sure how to modify it so that I can ...
2
votes
1answer
113 views
Why does the result hold for PIDs but not for UFDs?
Let $R$ be a subring of an integral domain $S$, and suppose $R$ is a PID. Then it follows that if $r\in R$ is a gcd of $r_1$ and $r_2$ in $R$, where $r_1$ and $r_2$ are not both zero, then $r$ is a ...
2
votes
2answers
760 views
Greatest common divisor in the Gaussian Integers
Let $a$ and $b$ be integers. Prove that their greatest common divisor in the ring of integers is the as their greatest common divisor in the ring of Gaussian Integers.
Ring of Gaussian Integers is:
...
0
votes
4answers
1k views
Show that every ideal of the ring $\mathbb Z$ is principal
Let $\mathbb Z$ be the ring of integers. The question asks to show that every ideal of $\mathbb Z$ is principal. I beg someone to help me because it is a new concept to me.
2
votes
1answer
117 views
Queries on proof that every PID is a factorisation domain
I'm reading a proof from C. Musili's Rings and Modules that every PID is a factorisation domain.
The author defines a factorisation domain as a commutative integral domain $R$ with a unit such that ...
8
votes
1answer
265 views
Are all subrings of the rationals Euclidean domains?
This is a purely recreational question -- I came up with it when setting an undergraduate example sheet.
Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
5
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3answers
478 views
Dedekind domain with a finite number of prime ideals is principal
I am reading a proof of this result that uses the Chinese Remainder Theorem on (the finite number of) prime ideals $P_i$. In order to apply CRT we should assume that the prime ideals are coprime, i.e. ...
2
votes
1answer
145 views
Number of ideals of a PID modulo an ideal
Let $R$ be a Principal Ideal Domain and $(a)\neq(0)$ an ideal of $R$. Prove $R/(a)$ has a finite number of ideals.
2
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1answer
109 views
How can we write this $\mathbb{Q}[x]$-module as a direct sum of cyclic $\mathbb{Q}[x]$-modules?
If $L$ is the submodule of $\mathbb{Q}[x]^{(3)}$ generated by $(2x-1,x,x),(x,x,x),(x+1,2x,x)$. How do we write $\mathbb{Q}[x]^{(3)}/L$ as a direct sum of cyclic modules?
0
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1answer
173 views
$R$ is PID, so $R/I$ is PID, and application on $\mathbb{Z}$ and $\mathbb{N}$
I'm supposed to show in a part of an exercise that if we have a ring $R$ that is a principal ideal domain, then for any ideal $I$ in $R$, $R/I$ will also be a PID.
So $I=(i)$ for some $i \in R$, and ...
5
votes
2answers
379 views
How to show that $R/I$ is Artinian when R is PID
I'm working through some of Hungerfords "Algebra", and having trouble with Excercise VIII 1.2.:
Show that if $I$ is a non-zero ideal in a principal ideal domain (PID) $R$, then the ring $R/I$ is ...
0
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0answers
396 views
How do I get this matrix in Smith Normal Form? And, is Smith Normal Form unique?
As part of a larger problem, I want to compute the Smith Normal Form of $xI-B$ over $\mathbb{Q}[x]$ where
$$
B=\begin{pmatrix} 5 & 2 & -8 & -8 \\ -6 & -3 & 8 & 8 \\ -3 & ...
3
votes
1answer
98 views
Do a matrix and its transpose have the same invariant factors over a PID?
I suspect this is true since it holds in the case over a field. But suppose $A\in M_{m\times n}(R)$ where $R$ is a PID. Does it still hold that $A$ and $A^{T}$ have the same invariant factors?
...
2
votes
2answers
104 views
Sub-module over the Fraction field of a PID
R is a PID with field of fractions K. $M \subseteq K$ is a fin. generated R-Submodule. I am trying to show M is in fact generated by one element.
2
votes
1answer
353 views
Norm-Euclidean rings?
For which integer $d$ is the ring $\mathbb{Z}[\sqrt{d}]$ norm-Euclidean?
Here I'm referring to $\mathbb{Z}[\sqrt{d}] = \{a + b\sqrt{d} : a,b \in \mathbb{Z}\}$, not the ring of integers of ...
1
vote
1answer
160 views
Can we consider set of all composite integers as an ideal? And if yes, why then Z is a PID?
In this wikipedia article it is said that set Z is a principal ideal domain, i.e. each one of its ideals can be generated by a single element.
But if we consider set C of all composite integer numbers ...
-1
votes
1answer
232 views
Isomorphism between set of all R-modules homomorphisms
Let $S$ be an $R$-module where $R$ is an integral domain and let $P = \langle p \rangle $ be a prime ideal.
Define:
$S_{P}=\{s \in S: p^{n} s=0 \ \textrm{for some natural }n\}$.
As usual, denote ...
5
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1answer
454 views
How many real quadratic number fields have the class number 1?
I know that in general the number of ideal classes are not 1, and that there are only 9 imaginary quadratic number fields which are principal ideal domains, i.e. $\mathbb(Q(\sqrt{-m}))$ where m is 1, ...
10
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1answer
279 views
Finitely generated modules over PID
Let $A$, $B$, $C$, and $D$ be finitely generated modules over a PID $R$ such that $A\oplus $ $B$ $\cong$ $C\oplus $ $D$ and $A\oplus $ $D$ $\cong$ $C\oplus $ $B$ . Prove that $A$ $\cong$ $C$ and $B$ ...
