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

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2
votes
1answer
147 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?
1
vote
1answer
268 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 ...
6
votes
2answers
600 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 ...
3
votes
1answer
722 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
130 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
122 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
676 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
180 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
vote
1answer
285 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 ...
2
votes
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
votes
1answer
592 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
votes
1answer
328 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 $B$ ...