Questions tagged [principal-ideal-domains]
For questions about principal ideal domains: rings without zero divisors where every ideal is principal.
664
questions
0
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3answers
25 views
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 ...
0
votes
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 ...
0
votes
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 $M$...
0
votes
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)
$M \...
0
votes
2answers
38 views
Use of “$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 ...
0
votes
0answers
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 $\forall x \in max(R)$? ...
-1
votes
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.
2
votes
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 ...
0
votes
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....
1
vote
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?
0
votes
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 ...
-1
votes
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 ...
-2
votes
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:
$$ M = 0.$$
1
vote
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}(...
4
votes
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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.
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
-1
votes
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
...
4
votes
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 ...
1
vote
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 $K$-...
-1
votes
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?
0
votes
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\...
0
votes
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^{\...
2
votes
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 ...
1
vote
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 ...
0
votes
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-...
1
vote
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 ...
1
vote
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 ...
3
votes
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=\...
0
votes
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 ...
0
votes
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?
0
votes
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{...
1
vote
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,...
1
vote
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 \...
0
votes
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^{...
2
votes
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=\...
0
votes
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 ...
3
votes
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 $...
1
vote
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) = \...
1
vote
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,{...
0
votes
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 ...
0
votes
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 ...
1
vote
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)...
