# Questions tagged [principal-ideal-domains]

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

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### R is UFD. R is PID if every prime ideal is principal. [duplicate]

Suppose not. We consider the set of all non-principal ideals, $S$. Order $S$ by inclusion. We show S satisfies all the conditions in Zorn's Lemma. So it has a maximal element. If we show the maximal ...
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### Why Smith normal form gives isomorphic modules?

I have an answer to the problem but I use some (trivial) diagram chasing by $5$-Lemma. Consider a principle ideal domain $A$ and a finitely generated module $M$ over $A$. Since $A$ is Noetherian, we ...
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### Prove every ideal in ring of integers is principal [duplicate]

I wrote a proof and would like to see how sturdy it is. I'm very new to this subject, and am curious. I wrote: We have $I \vartriangleleft Z$. Then for $x,y\in I$ and $z\in \mathbb{Z}$, $x - y \in I$, ...
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### If $I$ is an ideal of $R=\{\frac{a}{b}\in \mathbb{Q}\mid p\not\mid b\}$, for some fixed prime $p$, show that $(p^t)\subset I.$

Background: Exercise 14: Let $p$ be a fixed prime integer and let $R$ be the set of all rational numbers that can be written in the form $\frac{a}{b}$ with $b$ not divisible by $p$. Prove that (a) $R$ ...
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### Need help showing that the only submodules of $M$ are the ones in an ascending chain.

$\color{Green}{Background:}$ $\textbf{Assumed facts:}$ $\textbf{Theorem 1:}$ Let $R$ be a ring. Then the following conditions are equivalent: $(1)$ Every ideal of $R$ is finitely generated $(2)$ ...
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### Let R be a UFD such that each maximal ideal of R is a principal ideal, prove that R is a PID

I need help to the following problem: Problem: Let $R$ be a Unique Factorization Domain such that each maximal ideal of $R$ is a principal ideal. Then $R$ is a Principal Ideal Domain. Solution(my ...
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1 vote
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### $\mathbb Z[\xi_{2n}]$ is a pid

Let $\xi_{2n} \in \mathbb C$ a primitive $2n^{th}$ root of unity for some integer $n\ge 2$. Is the inclusion $\mathbb Z[\xi_{2n}] \hookrightarrow \mathbb C$ flat? It is possible to answer this ...
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### Deduce Jordan Normal Form from PID finitely generated module structure.

Let $k$ be an algebraically closed field. Any $k$-linear map $\phi:k^n\to k^n$ imposes an extra $k[x]$-module structure on $k^n$ by defining $p(x)\cdot v = p(\phi)(v).$ Clearly then $k^n$ is a ...
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### Free submodules of an integral $R.$

I was told by the author of the answer here Showing that the rank of $M$ is exactly $1.$ that: Free submodules of an integral domain $R$ are exactly the principal ideals of $R.$ I am wondering which ...
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### Pure submodule that is a direct summand

I have been recently studying about Pure submodules of a module in a commutative algebra course. I came up with the following two questions, please help me as I am completely stuck. The definition of ...
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### Primary component of a finitely generated module over PID is a direct sum of cyclic modules.

Recently I have been studying the structure theorem of finitely generated modules over a principal ideal domain $R$. I am stuck with the following statement and I am looking for an elegant and nice ...
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### If every irreducible elements in Noetherian ring is prime, it is UFD

I have problem with the proof of theorem 4 in this link It says if $R$ is an Noetherian ring, we construct $\mathbb U$, the set of ideals generated by each element of $R$ that cannot be written as a ...
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### Exhibit all the ideals in the ring $F[x]/(p(x))$, where $F$ is a field and $p(x)∈F[x]$ (describe them in terms of the factorization of $p(x)$)

This is Exercise 9.2.5 in Dummit and Foote's Abstract Algebra Exhibit all the ideals in the ring $F[x]/(p(x))$, where $F$ is a field and $p(x)$ is a polynomial in $F[x]$ (describe them in terms of ...
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### What is meant by "invertible" matrices in the creation of a SNM

I just read up on wikipedia on the Smith Normal Matrix. But what is meant by an invertable matrix. For example if you have a start matrix with only PID values does that mean the other matrices don't ...
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### Are the invertible matrices which are used to find the SNF always part of the ideal principle domain?

Let's say SNF = TAT^-1. Do T and T inverse always only have elements part of the domain? For example, if we have an integer matrix, will T and T inverse only have integer values (not rational numbers)....
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### $P$ is a prime ideal $\iff$ for $r, s \in R$ such that $rRs \subset P$, then $r \in P$ or $s \in P$.
If $P$ is an ideal in a not necessarily commutative ring $R$, then the following conditions are equivalent: a) $P$ is a prime ideal, b) If $r,s\in R$ such that $rRs\subset P$ then $r\in P$ or $s\in P$...