# Questions tagged [principal-ideal-domains]

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

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### Finding all the maximal ideals of $\mathbb{Z}_{63}$

How do I go about finding all the maximal ideals of this ring ? I realise that all ideals are subgroups with respect to addition. Therefore, since $\mathbb{Z}_{63}$ is cyclic then every subgroup, and ...
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### If $a+bc$ and $c$ have irreducible factors in common, then $a$ and $c$ have the same irreducible factors in common.

Let $a,b,c \in K[t]$ where $K$ is field with characteristic not $2$ or $3$ and see title for the question. This is a problem I encountered in showing the image of some map is finite, which I need in ...
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### Applying the elementary divisor theorem

I've just started studying this topic and I've stopped at this exercise: "Let $M = \mathbb{Z}^3$ and $N$ the submodule generated by $\{(1,1,6),(1,-1,6)\}$. Determine a basis of $\{v_1,v_2,v_3\}$ of $M$...
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### Isn't $I$ a maximal ideal in $\Bbb Z_{11} [X]$?

Consider the ideal $I$ defined by $$I : = \left \{ f(x) \in \Bbb Z_{11}[X]\ :\ f(2) = 0 \right \}$$ in $\Bbb Z_{11}[X].$ Is $I$ a maximal ideal in $\Bbb Z_{11} [X]$? My attempt $:$ What I think is ...
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### Given a nonconstant $f\in \Bbb C[x]$, condition for every finitely generated indecomposable $\Bbb C[x]/(f)$-module is projective

Let $f$ be a nonconstant polynomial with complex coefficients, and consider the ring $R = \Bbb C[x]/(f)$. I am trying to prove that $R$ has no nonzero nilpotent if and only if every finitely ...
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### Does being principal ideal ring with identity implies PID?

I have recently stumbled upon a standard proof that if $R$ is a Euclidean ring, then $R$ is a PID. But in the proof they first show that $R$ is a principal ideal ring with identity. But then they ...
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### Cyclic Modules over a PID

Let $R$ be a PID and $M$ be an $R$-module. If $M$ is finitely generated then show that $M$ is cyclic if and only if $M/PM$ is cyclic for every prime ideal $P$ of $R$. Show that the ...
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### Question about integer number and PID

What is the set of all integers that can be written in the form $m^2+2n^2$? Here $m$, $n\in \mathbb{Z}$. Further more, what about $m^2+kn^2$ for some $k\in \mathbb{Z}^{+}$? I think the first question ...
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### Show that any non zero ideal of a principal ideal domain is a unique product of prime ideals

I know that in a PID all the ideals are principal ideals but how to connect this with the given question? I have no idea how to prove this.Any help will be highly appreciated.
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### Find necessary and sufficent condition question

In P.I.D $K$ give 2 elements $a,b$. Find necessary and sufficent condition that: "If $J$ is an ideal of $K$ that $(a) \subseteq J \subseteq (b) \text{ then } J=(a) \text{ or } J=(b)$". My attempt: I ...
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### If $R$ is a PID, $S$ an integral domain and $f: R \to S$ is an epimorphism, why is it that either $f$ is an isomorphism or $S$ is a field?

If $R$ is a PID, $S$ an integral domain and $f: R \to S$ is an epimorphism, why is it that either $f$ is an isomorphism or $S$ is a field? PID - Principal Ideal Domain What I know: If $S$ is not a ...
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### If $v$ is a discrete valuation on a field $K$, then $A:=\left\{ a \in K : v(a) \geq 0 \right\}$ is a principal ideal domain.

J.S. Milne's Algebraic Number Theory says [Proposition 3.27] Let $v$ be a discrete valuation on $K$, then $$A:=\left\{ a \in K : v(a) \geq 0 \right\}$$ is a principal ideal domain with maximal ...
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### If $R$ is a PID and $p$ is a prime element, prove that $R/\langle p^k\rangle$ is a strongly associate ring for any positive integer $k$.

Question: If $R$ is a PID and $p$ is a prime element, prove that $R/\langle p^k\rangle$ is a strongly associate ring for any positive integer $k$. I saw this question somewhere before but the answer ...
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### In a PID, every irreducible element is a prime element. What's wrong with following?

if $p=ab$ $\implies p|ab$ If $p$ is irreducible then either $a$ or $b$ is a unit. If $a$ is a unit, then $a^{-1}p=b$ or, $b \in <p>$ $\implies b=pt \implies p|b$ Thus $p$ is prime ...
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### Non-integer domain which every ideal is a principal ideal

Let $F$ be field and $A=F[t]\setminus (t^2)$, where $(t^2)$ is the ideal of $F[t]$ (a) Show that every ideal of $A$ is principal ideal (b) Find all prime ideals of $A$ I know $A$ is not integer ...
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### Why is the largest invariant factor the minimal polynomial, and why is the product of invariant factors the characteristic polynomial?

I'm just learning the primary decomposition theorem for finitely generated modules over a PID, and its application to linear algebra. Let $V$ be a vector space over $K$, and let $T: V \to V$ be a $K$-...
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### How does an ideal generated by $(x^{2})$ in $Z[x]$? [closed]

I was just wondering whether there would exist an element $x^{-2}$ and how would the elements in general look like?
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### Finitely generated module over PID with tensor with itself is zero

Hello I have the next doubt about this problem: Show that if $A$ is a finitely generated module over a PID and $A\otimes_{\Lambda}A=0$, then $A=0$. I have done the next thing, I consider the next ...
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### Invertible operator such that its characteristic and minimal polynomials coincide

Let $T \in GL(V) \subset End(V)$, where $V$ is finitely dimensional vector space over field. Suppose the minimal polynomial $\mathcal{m}_T$ and it's characteristic polynomial $\chi_T$ coincide. What ...
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### Polynomial Ring $\mathbb{C}[X,Y]$

Consider the polynomial ring $\mathbb{C}[X,Y]$ and the ideal $I$ generated by $Y^2-X$. How many maximal ideals are there in Quotient ring $\displaystyle\frac{\mathbb{C}[X,Y]}{I}$ solution i tried-...
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### Factorization in a principal ideal ring/rng

It is known that every PID is a UFD. Is it true that every element of a commutative principal ideal ring (PIR) or rng that is not zero and not a unit is a product of a finite number of irreducible ...
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### Finitely generated modules over PIDs and vector spaces

I am trying to understand and complete the proof of the following statement: Let $A$ be a module over the ring $\mathbb{C}[x]$. Since $\mathbb{C}\subset\mathbb{C}[x]$, $A$ is a $\mathbb{C}$-vector ...
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### Bogus proof that every ideal in a Dedekind domain is principal

Let $A$ be a Dedekind domain and $I$ a nonzero ideal of $A$. For every $a \in I$, $(a)$ is contained in $I$, so $I$ divides $(a)$ and there exists some ideal $J_a$ such that $(a)=IJ_a$. We have I=\...
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### Tensoring with fraction fields kills the torsion

Assume $R$ is a PID. And $M$ is finitely generated $R$-module. So we have the classification theorem: $M\cong R^r\oplus T(M)$. Is it true that $M\otimes_{R}\mathrm{Frac}(R)=\mathrm{Frac}(R)^r$? At ...
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### Uniqueness of generator of Principal ideal domain

Just this simple question, I don't know why I am stuck: Is the generator of a Principal ideal domain unique?
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### Homomorphisms between $R/(p_1^{e_1})$ and $R/(p_2^{e_2})$ when $p_1$ and $p_2$ are associated.

I have given that $R$ is a PID, $p_1,p_2 \in R$ are associate prime elements and $e_1,e_2 \in \mathbb{Z}_{>0}$. I now have to show that \begin{equation} \mathrm{Hom}_R\big(R/(p_1^{e_1}), R/(p_2^{...
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