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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1answer
28 views

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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0answers
34 views

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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1answer
21 views

Question about a step in the proof of the uniqueness of the decomposition of a finitely generated R-module, R a PID.

The theorem is the following: Let $R$ be a principle ideal domain, and let $M$ be a finitely generated $R$-module. Suppose that $M \cong R^s \bigoplus R/Ra_1 \bigoplus ... \bigoplus R/Ra_u$ (1) $M \...
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2answers
38 views

Use of “$A$ is a domain” in the proof that $Q$ is an injective $A$-module iff it is divisible

Let $A$ be a PID. Then, an $A$-module $Q$ is injective iff $Q=rQ$ for every $r\neq 0$ in $A$. My question is, where is the property "A is a domain" used in the proof of the above? Can someone please ...
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34 views

Counter-example Modules

If we're given an arbitrary principal ideal domain $R$ and some $R$-module, call it $M$, is there an example of an $M$ that is not injective while $M_x$ is an injective module $\forall x \in max(R)$? ...
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0answers
34 views

Length of quotient of ring and ideal in a PID of A-module

I am trying to find the length of in a PID $A$ of the quotient $\frac{A}{I}$ where I is an ideal of $A$. I don't know how to approach it, thanks.
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4answers
83 views

Why is the polynomial ring $\Bbb R[x]$ a PID but $\Bbb Z[x]$ is not?

Why is the polynomial ring $\Bbb R[x]$ a PID but $\Bbb Z[x]$ is not? The question is asking me to prove that $\Bbb R[x]$ is a PID. I'm assuming you would go about this knowing that every field is a ...
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0answers
19 views

Show a multivariate polynomial ring is not a PID if it contains 1. [duplicate]

I have written a proof for the following statement and am unsure if it is valid. Let $R$ be a commutative ring with $1$. Prove that a polynomial ring in more than one variable over $R$ is not a PID....
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0answers
30 views

Intersection of two principal ideals in an integral domain. [duplicate]

Could anyone give a specific example where the intersection of two principal ideals in an integral domain is not a principal ideal?
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2answers
32 views

Let $R$ be a PID, and let $\pi\in R$ be prime. Is it $R/\langle\pi\rangle \times R/\langle \pi\rangle$ a cyclic $R$-module?

Need help with this question: Let $R$ be a PID, and let $\pi\in R$ be prime. Is it ever the case that $$R/\langle\pi\rangle \times R/\langle \pi\rangle$$ is cyclic as $R$-module? So my attempt ...
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1answer
63 views

Is $\Bbb Z / n \Bbb Z$ a PID? [closed]

I’d like to when $\Bbb Z / n \Bbb Z$ is a PID. I don’t know if depends of the value of $n$, it is true for all $n$ or $\Bbb Z / n \Bbb Z$ is never a PID. No idea. In case it’s true I’d like to see the ...
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1answer
60 views

Let $A$ be a PID, $M$ an injective finitely generated module. Prove that $M = 0$. [closed]

Help! Let $A$ be a PID, $M$ an injective finitely generated module. Prove that: $$ M = 0.$$
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0answers
39 views

In what rings is every ideal a fractional ideal?

My instructor has said that, if $K$ is a number field, and $O_K$ is its ring of integers, any ideal of $O_K$ is a fractional ideal. This is what allows us to say that, if the class group $\mathrm{Cl}(...
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3answers
73 views

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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1answer
48 views

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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1answer
42 views

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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1answer
50 views

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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1answer
59 views

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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1answer
29 views

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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1answer
44 views

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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2answers
58 views

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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1answer
30 views

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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0answers
28 views

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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1answer
36 views

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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2answers
43 views

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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0answers
40 views

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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2answers
32 views

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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1answer
34 views

Is the ideal generated by $a$ simply $\{ au\mid u\in R, a\in I \}$ ($R$ is a ring)

More specifically, $R$ is a commutative ring. I'm trying to understand what the ideal "generated by $a$" is, where $a$ is an element of $R$. I believe this ideal is simply the set $\{a\cdot u\mid u\...
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1answer
28 views

Diophantine in Q[x]

Suppose $f, g \in Q[x]$ non-zero elements. Let $(f, g) = (d),$ and $h \in (d).$ Then there exit polynomials $p,q$ such that $h =pf + gf.$ I want to show that $h= p^{\prime}f +q^{\prime}g$ iff $p^{\...
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1answer
46 views

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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1answer
38 views

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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1answer
59 views

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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0answers
80 views

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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0answers
31 views

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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2answers
57 views

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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1answer
72 views

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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1answer
41 views

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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1answer
51 views

Prove that an ideal of a subring of $\mathbb{Q}$ is principal

Let $R$ be a subring of $\mathbb{Q}$ containing $1$. We denote the absolute value of an integer $n$ by $|n|$. Let $I$ denote a non-zero ideal of $R$. Let $0\ne b,a\in \mathbb{Z}$ be such that $\frac{...
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1answer
84 views

Let R be a principal ideal ring, prove R is a Noetherian ring.

Let R be a principal ideal ring, prove R is a Noetherian ring. know we have to construct an ascending chain of principal ideals in R. And take their union, this is obviously an ideal. Since R is a PID,...
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1answer
65 views

Module in exact sequence isomorphism

I'm really struggling with this excercise although it does not seem that hard. We are given a PID $R$ and a prime element $p \in R$. $M$ is an $R$-module and a short exact sequence is given by $0 \...
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1answer
26 views

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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2answers
121 views

Number of ideals of norm $100$ of the Kleinian integers

The question to begin with is: how many ideals of norm $100$ does the ring of integers of $K=\mathbb{Q}(\sqrt{-7})$ have? I know the following stuff for certain: $\Delta_K=-7$ and $\mathscr{O}_K=\...
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1answer
36 views

Proof: PID is UFD: not understanding step

One step of the proof is the following. Let $R$ be a PID. Let $x$ be a non-zero, non-unit element in $R$. Suppose $x$ can not be written as a product of irreducible elements, then $x$ is not ...
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1answer
60 views

$R$ is not a principal ideal domain (show)

Let $R=\mathbb{Z}[\sqrt{-17}]=\lbrace m+n\sqrt{-17}|m,n \in \mathbb{Z} \rbrace$. How to show that $R$ is not a principal ideal domain? My way was: Let $I \subsetneq R$ be an ideal of $R$, given by $...
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2answers
60 views

Suppose $a$ is odd and $b$ is even. Prove that $\operatorname{gcd}(a,b) = \operatorname{gcd}(a+b, a-b)$. [duplicate]

This question originates from Pinter's Abstract Algebra, Chapter 22 Exercise E1. Let $a$ and $b$ be integers. Suppose $a$ is odd and $b$ is even. Prove that $\operatorname{gcd}(a,b) = \...
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2answers
74 views

Constructing invertible matrix with entries in a principal ideal domain

Am trying to solve this exercise in S. Bosch's book Algebra. From the Viewpoint of Galois Theory (Exercise 5, page 81). Assume that $A$ is a principal ideal domain, and assume that $({a_{11}},\dots,{...
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1answer
59 views

If $A$ is a DVR such that $k\subset A\subset k(x)$, $Frac(A)=k(x)$ and $x\not\in A$, then $A=\mathcal{O}_{\infty}(k)$

Let $k$ be an algebraically closed field. Suppose we have a discrete valuation subring $A$ of $k(x)$ containing $k$ and whose quotient field is exactly $Frac(A)=k(x)$. If we are given that $x\not\in ...
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1answer
35 views

Prime and maximal ideals of $\mathbb{C}[x,y]$.

Do the set $\{ p(x,y): P(a,b)=0\}$ is a maximal( or prime) ideal of $\mathbb{C}[x,y]$. If so what will be its principal ideal form representation.(Since all ideals are principal). I feel the ideal ...
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1answer
23 views

If $M=D/(d)$ then $M=t_{p}(M)\oplus t_{q}(M)$

I am interested in the following result: Let $D$ be a $PID$. Let $d\in D$ such that $d=p^{r}q^{s}$ with $s,r\geq 1$ where $p,q$ are non-associated irreducible elements. If $M=D/(d)$ then $M=t_{p}(M)...

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