# Questions tagged [principal-ideal-domains]

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

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### How can I finish the proof for the following proposition: every ideal in k[x] is a principal ideal? [duplicate]

Proposition : Every ideal in k[x] (polynomial ring) is a principal ideal Proof : Suppose that $I\subseteq K[x]$. Take $p(x)\in I$ such that $p(x)$ is monic and $deg(p(x))$ is minimal over all ...
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### On Showing that $\mathbb{Z}[x]$ is not a Principal Ideal Domain

I'm working on a question that asks to consider the ring $R = \mathbb{Z}[x]$. I need to show that $(x)$ is a prime ideal, show that $(x,7)$ is a prime ideal and use these facts to conclude that $R$ ...
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### Question about the relation between P.I.D. and quotient field. [duplicate]

Let $D$ be a P.I.D. and $F$ is a quotient field of $D$. How to prove that any subring $D'$ contained in $D$ in $F$ is a P.I.D., and $D'$ is a ring of fraction of a subset of $D$ closed under ...
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### Question on Definition of Canonical Epimorphism

Let $P$ be a finitely-generated $R$-module, where $R$ is a principal ideal domain. The text that I am reading refers to $$f:\bigoplus_{p \in P}R \rightarrow P$$ as the canonical epimorphism of $R$-...
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### How to Prove an Ideal Can Be Generated From 2 Elements [duplicate]

Given a commutative ring with unity, $R$, an $a,b \in R$, and an ideal $I$ such that $I=\{ax+by \mid x,y \in R\}$. Prove that $I=(a,b)$. I think I want to show that $R$ is a PID, but I am not quite ...
1 vote
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### What does the correspondence theorem tell us about the ideals of $\Bbb Z[x]$ that contain $x^2+1$?

What does the correspondence theorem tell us about the ideals of $\Bbb Z[x]$ that contain $x^2+1$? If I define $\varphi : \Bbb Z[x] \to \Bbb Z[i]$ as $x \longmapsto i$, then $\ker \varphi = (x^2+1)$. ...
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### Why are the direct summands obtained in the structure theorem for finitely generated modules over a PID indecomposable?

The structure theorem tells us that a finitely generated module $M$ over a principal ideal domain $R$ is isomorphic to a direct sum $\bigoplus _{i}R/(q_{i})$ where $( q_i ) ≠ R (q_{i})\neq R$ and ...
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### How does this quotient by a maximal ideal looks like?

I have the following situation: Let $R$ be a PID. Then let $\mathfrak{p}\subset R$ be a maximal ideal. We consider the set $$\left(R/\mathfrak{p}\right)[X]$$ which is clearly a quotient, but I would ...
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### Content of a polynomial and maximal ideals.

I have the following problem: Let $R$ be a PID. Let us denote for $P=a_0+a_1 X+\dots+a_n X^n\in R[X]$ that $c(P)=\gcd(a_0,...,a_n)$. Show that for $P\in R[X]$ nonzero $c(P)=1$ iff for every maximal ...
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### Equivalent condition for uniserial torsion module over a PID

This is a homework problem. Suppose, M is a finitely generated torsion module over a PID R. Then, M is uniserial if and only if M has only one elementary divisor. So far I was able to prove the ...
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### Checking if given polynomials are units in $\mathbb{Z}_7[x]$ [duplicate]

So, I was doing an algebra exercise related to $\gcd$'s and $PID$'s and I need to check if some polynomials in $\mathbb{Z}_7[x]$ are units. The polynomials are the following: \begin{equation*} g = x^2+...
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### Chinese Remainder Theorem for Hurwitz quaternions

I know that if we have a noncommutative ring the CRT (Chinese Remainder Theorem) doesn't work and I know that the CRT works for all PIDs (Principal Ideal Domains). My question is: In the case of the ...
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### Hypothesis of Structure Theorem for Finitely-Generated Modules over PID's [duplicate]

As a consequence of the Structure Theorem for Finitely Generated Modules over a PID, we know that if $R$ is a PID, and $M$ is an $R$-module, then $M$ is a direct sum of its torsion submodule $T(M)$ ...
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### Finitely generated torsion module homomorphism in PID local ring

Problem Statement: Let R be a commutative local ring, PID, but not a field. Given two nonzero finitely generated torsion $R$-modules, show that there exists a nonzero $R$-module homomorphism $M \to N$....
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### Hypothesis - am I doing this right? (Local, principal ideal ring)

Let $R$ a commutative local principal ideal ring with 1 that is not Artinian. So it's Krull dimension is non zero. Let $P\subsetneq M$ a prime ideal and M the maximal ideal of R, since P and M are ...
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