For questions about principle ideal domains: rings without zero divisors where every ideal is principle.

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1answer
97 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 = ...
14
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2answers
976 views

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

Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID? Any ...
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1answer
238 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 ...
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1answer
1k 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 ...
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1answer
127 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
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2answers
1k 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: ...
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4answers
3k 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.
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1answer
138 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 ...
10
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1answer
371 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 ...
6
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3answers
814 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. ...
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1answer
585 views

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

I read this proof that if $D$ is an integral domain and $D[X]$ is a principal ideal domain, then $D$ is a field. My question is if the requirements can be relaxed a bit, namely: Is it true that ...
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1answer
191 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
155 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?
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1answer
294 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 ...
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2answers
649 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 ...
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1answer
806 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
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1answer
132 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
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2answers
125 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
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1answer
742 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 ...
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1answer
187 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
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1answer
289 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 ...
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6answers
2k views

Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain

Please help me prove that the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain. This was from Abstract Algebra
5
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1answer
630 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
342 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 $B$ ...