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

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Inverse image of a PID is a PID

Let $f : R \to S$ be a ring homomorphism from $R$ onto $S$. If $S$ is a PID, is $R$ then a PID? If this is not possbile, is there an example to contradict it?
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47 views

Invariant factors of torsion module under homomorphism

Let M be a torsion module for a PID D with invariant factors $(d_1) \supseteq (d_2) \supseteq...\supseteq (d_r) $ (which means $M = Dz_1 \bigoplus Dz_2 \bigoplus ... \bigoplus Dz_r $ s.t $ann(z_i) ...
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0answers
37 views

Quotient of a PID by a prime ideal [duplicate]

Prove that quotient of a PID by a prime ideal is PID.
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2answers
277 views

Specific way of showing $\Bbb Z[\sqrt{-d}]$ is not a Euclidean Domain when $d>2$

Is it true that if a ring is not a UFD then it's not a Euclidean Domain? I have a ring $R=\mathbb{Z}[\sqrt{-d}]=\{ a+b\sqrt{-d} \mid a,b \in \mathbb{Z} \}$ where $d$ is a square free integer. I want ...
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1answer
63 views

Describe units and maximal ideals in these two PIDs

If $p$ is a fixed prime integer, let $R$ be the set of all rational numbers that can be written in the form $(a)$ $\frac{a}{b}$ with $b$ not divisible by $p$. $(b)$ $\frac{a}{b}$ with $b=p^k$ for a ...
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3answers
67 views

Showing that an integral domain is a PID if it satisfies two conditions

This is just a textbook problem from Dummit and Foote, but the issue is that our class barely touched on PIDs and the preceding material, so I don't really know or understand much. Anyway, Let ...
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1answer
43 views

gcd & lcm in a PID

In a PID, $l={\rm lcm}(a,b)$ and $d=\gcd(a,b)$. Is it always true that the following product ideals are equal? $$<d><l> = <a><b>$$ Thanks in advance -- Mike
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2answers
57 views

Why $\langle I, J\rangle =R$ for distinct prime ideals $I$, $J$ of a principal ideal domain $R$?

Let $R$ be a principal ideal domain with identity and $I$, $J$ be distinct prime ideals of $R$. Prove that $1 \in \langle I, J\rangle$ hence $\langle I, J\rangle = R$. How to prove?
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90 views

Subring of the field of rational numbers

Let $R=\{a\cdot2^n\mid a,n \in \mathbb{Z}\}$ be a subring or the field of rational numbers $\mathbb Q$. i) What kind of elements are invertible in $R$? ii) Prove that $R$ is a principal ...
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1answer
103 views

Quotient ring of $\Bbb Z[x]$ by an irreducible polynomial is a PID

I don't know what can I do with this problem. How can I prove that $\mathbb{Z}[x]/(x^{3}-4x+2)$ is PID?
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3answers
217 views

$\mathbb Z\times\mathbb Z$ is principal but is not a PID

I need to find an example of a ring that is not a PID but every ideal is principal. I know that $\mathbb Z\times\mathbb Z$ is not an integral domain, so certainly is not a PID, but here every ideal is ...
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1answer
39 views

Generator for the ideal $I + J$ where $I = (2 + 3i)$ and $J = (1 - i)$

On a related question I calculated the GCD of $I = (2 + 3i)$ and $J = (1 - i)$ to be $1$. Now I know that $\mathbb{Z}[i]$ is a principal ideal domain. And I also know that the greatest common divisor ...
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2answers
164 views

surjective homomorphism between principal ideal rings

I asked this question before but did not get an answer, I tried solving it myself and I think I'm heading somewhere, I just need a push in the right direction. Let $\phi: R \to S$ be a homomorphism ...
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1answer
78 views

Ring homomorphism, maximal ideals

Here's a question from my worksheet, i solved subquestion (1) but can use help with the other 2...And also would appreciate any comments on my answer for subquestion (1). Let $\psi: R->S$ be a ...
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0answers
30 views

Inclusion-minimality of a lattice basis

An integer lattice is a subgroup of $\mathbb{Z}^n$. Since $\mathbb{Z}$ is PID, each lattice has a well-defined rank and a generating set of rank many elements is a basis. I wonder if there is a way ...
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2answers
798 views

Proving that a ring is not a Principal Ideal Domain

This is my first question on StackExchange. I'm taking a second semester course of Abstract Algebra. I have a general understanding of Principal Ideal Domains, but I am a bit confused on proving that ...
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1answer
53 views

Understanding this basic lemma proof in Ireland & Rosen

Lemma 2. Let $R$ be a PID and $p$ a prime element and $a \neq 0$ any element. Then there is an integer $n$ such that $p^n \ | \ a$ but $p^{n+1}$ doesn't divide $a$. Proof. If the lemma were ...
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1answer
26 views

In the proof that in a PID, every non-zero non-unit is the product of irreducibles…

In proving that all non-zero non-units of a PID are a product of irreducibles, theres: "We now show that $a$ is a product of irreducibles. If $a$ is irreducible, we are done. Otherwise let $p_1$ ...
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2answers
154 views

Integral domain in complex numbers.

Let $I = _{\mathbb {C} [X]} \langle X^2 + 1\rangle$ the principal ideal of $\mathbb{C}[X]$ generated by $X^2 + 1$. Is $\mathbb{C}[X]/I$ an integral domain? From my understanding $\mathbb{C}[X]/I$ ...
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68 views

Confusion regarding the proof of “Every PID satisfies the Ascending Chain Condition”.

I refer to this proof of the fact that Principal Ideal Domains satisfy the Ascending Chain Condition. It says Let $\bigcup\limits_{i=1}^{\infty}I_i=(a)$. As $a$ is present in ...
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1answer
60 views

Number of ideals in $\Bbb Z[x]/(x^3+1, 7)$

I am trying to find the number of ideals in $R:=\Bbb Z[x]/(x^3+1, 7)$ and $S:=\Bbb Z[x]/(x^3+1, 3)$. I started with $R$ and tried to write it in terms of familiar rings, by using fundamental ...
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1answer
97 views

PID modulo a non-zero ideal is a semilocal ring

Let $R$ be a commutative ring, $\mathfrak{m}\subset R$ a maximal ideal and $f$ a monic polynomial in $R[x]$. I want to show that $A:=\frac{R[x]}{\mathfrak{m}[x]+(f)}$ is a semilocal ring, where ...
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1answer
70 views

Counting the ideals of $\frac{\mathbb{R}[X]}{(X^2)}$

I want to ask you guys if I'm on the right track: Here's the question: Suppose $a \in \mathbb{R}$. Count the ideals of $\frac{\mathbb{R}[X]}{(X^2-a)}$. Give an example of a ring with exactly 3 prime ...
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1answer
117 views

number of element in a principal ideal domain can be $25/36/35/15$?

Could any one tell me number of element in a principal ideal domain can be $25/36/35/15$ ? I just know a principal ideal domain is generated by a single element. what the knowledge I need to find ...
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1answer
106 views

Principal Ideal Domain Basics.

Let $R$ be a Principal Ideal Domain and $a,b,c,d$ elements in $R$, such that $ab-cd=1$. I am trying to figure out why $Rb \cap Rd=Rdab+Rbcd$. In case this is true, I am wondering weather it is enough ...
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1answer
46 views

Ideals of nested PID's

Let $R\subset K$ be principal ideal domains. If $a,b$ are nonzero elements of $R$, prove that $I=J\cap R$, where $I$ and $J$ denote the ideals generated by $a,b$ in $R$ and $K$, respectively. Showing ...
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0answers
50 views

Is quantum torus a principal ideal domain?

For a quantum torus $C_q[x_1^{\pm1}, ...,x_n^{\pm1}]$ satisfying $x_ix_j=q_{ij}x_jx_i$. Question: Is this quantum torus a principal ideal domain?
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1answer
135 views

Find a non free submodule of a free module over R which is not PID

I try to solve following question. Show that $R=\mathbf{Q}[x,x^{-1}]$ is not a PID. Construct a free module over $R$ having a non free submodule. One may give some examples for free modules but not ...
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265 views

Why define vector spaces over fields instead of a PID?

In my few years of studying abstract algebra I've always seen vector spaces over fields, rather than other weaker structures. What are the differences of having a vector space (or whatever the ...
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2answers
235 views

radical of sum of two ideals

$I$ and $J$ are ideals in $k[x_1,\cdots,x_n]$. Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$. I have no idea how to prove it. Can someone help?
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2answers
121 views

Show that if $D$ is a domain but not a field then $D[x]$ is not a principal ideal domain. [duplicate]

Show that if $D$ is a domain but not a field then $D[x]$ is not a principal ideal domain. Sorry for my english....
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3answers
143 views

Show that $\mathbb{Z}[x]=\lbrace \sum_{i=0}^{n}{a_ix^i}:a_i \in \mathbb{Z}, n \geq 0 \rbrace$ is not a principal ideal ring.

Show that $\mathbb{Z}[x]=\lbrace \sum_{i=0}^{n}{a_ix^i}:a_i \in \mathbb{Z}, n \geq 0 \rbrace$ is not a principal ideal ring. I know the definition of principal ideal ring is that every ideal is ...
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1answer
114 views

which of the following statements are true and why?

which of the following statements are true and why? Any two irreducibles in any UFD are associates. If $D$ is a PID, then $D[x]$ is a PID. In any UFD, if $p|a$ for an irreducible $p$, then $p$ ...
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1answer
391 views

Domain of a complex function

1) Why do the domain of a complex function has to be a disk (circular neighborhood of Zo)? $$ |z-z_0|<p $$ 2) Domain is an open connected set. An open set D is said to be connected if every pair of ...
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413 views

Show that $\mathbb{Z}[\theta]$ (where $\theta = (1 + \sqrt{19}i)/2$) is a principal ideal domain.

I'm having difficulties with a homework problem from Algebra by Hungerford. Let $R$ be the following subring of the complex numbers: $R = \{a + b(1 + \sqrt{19}i)/2 \mid a, b \in \mathbb{Z}\}$. ...
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3answers
733 views

Ring of trigonometric functions with real coefficients

Let $R$ be the ring of functions that are polynomials in $\cos t$ and $\sin t$ with real coefficients. Prove that $R$ is isomorphic to $\mathbb R[x,y]/(x^2+y^2-1)$. Prove that $R$ is not a unique ...
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1answer
154 views

Units in $\mathbb{Z}[\sqrt[3]{2}]$ : $\pm(1+\sqrt[3]{2}+(\sqrt[3]{2})^2)^n$?

The subring $\mathbb{Z}[\sqrt[3]{2}]\subset\mathbb{C}$ is a PID. I remember reading somewhere that the units in $\mathbb{Z}[\sqrt[3]{2}]$ are precisely the elements ...
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1answer
162 views

Quotient of ring of integers

Let $R=\mathcal{O}(K)$ be the ring of the integers of $K=\mathbb{Q}[\zeta_8]$, where $\zeta_8=e^{2\pi i/8}=\sqrt{2}/2(1+i)$ is a primitive eighth root of unity in $\mathbb{C}$. It can be shown that ...
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1answer
52 views

“Two ways” to reduce a module

Let $M$ be a module over a principal ideal domain $R$ and $\mathfrak{m}$ a maximal ideal of $R$ with residue field $R/\mathfrak{m}=k$ of characteristic $p$. Under what circumstances are the modules ...
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5answers
512 views

How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
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1answer
83 views

Reference on Operators on Modules over PID

The operators on finite dimensional vector spaces over any field can be seen in nice form w.r.t. some choice of basis such as "diagonal, triangular, Jordan form, rational form etc." But, for ...
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1answer
116 views

Is $(x^3-x^2+2x-1)$ prime in $\mathbb{Z}/(3)[x]$?

This is somewhat of a follow up on this question: Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}[x]$? I'm curious, is $\mathbb{Z}[x]/I$ a domain, with $I=(3,x^3-x^2+2x-1)$? I know $I$ is not ...
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0answers
335 views

Exercise on modules over PID involving injective modules, Baer's criterion.

I'm interested if I solved this somewhat correctly, and would like to be set straight if it is wrong. This is an exercise from an introductory text. Let $A$ be a module over a principal ideal ...
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2answers
1k views

Every prime ideal is either zero or maximal in a PID.

$(1)$ Let $R$ be a commutative ring with $1\neq 0.$ If $R$ is a PID, show that every prime ideal is either zero or maximal. In many books I have found the proof of the above statement where they ...
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4answers
888 views

An integral domain whose every prime ideal is principal is a PID

Does anyone has a simple proof of the following fact: An integral domain whose every prime ideal is principal is a principal ideal domain (PID).
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1answer
186 views

Question about torsion submodules and Decomposition theorem

Let $A$ be an principal ideal domain, and $M$ an $A$-module. If $p$ is irreducible in $A$, let's define $$\mathrm{Tor}_p(M):=\{m\in M\mid p^km=0\text{ for some }k\in\mathbb{N}\}.$$ I need to show ...
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1answer
175 views

Integer extensions, rings $\mathbb{Z}[\sqrt{s}]$

I'm not sure if this type of question is acceptable here, but I'd really appreciate someone's help. I'm about to start writing a semestral work that we need to achieve the Bachelor's degree in our ...
3
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1answer
1k views

Submodule of free module over a p.i.d. is free even when the module is not finitely generated?

I have heard that any submodule of a free module over a p.i.d. is free. I can prove this for finitely generated modules over a p.i.d. But the proof involves induction on the number of generators, so ...
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2answers
252 views

A question about Euclidean Domain

This is a problem from Aluffi's book, chapter V 2.17. "Let $R$ be a Euclidean Domain that is not a field. Prove that there exists a nonzero, nonunit element $c$ in $R$ such that $\forall a \in R$, ...
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1answer
198 views

Ring of analytic functions on the circle

Let $A = C^\omega(S^1)$ (resp. $C^\omega_{\mathbb C}(S^1)$) the ring of real-analytic real-valued (resp. complex valued) functions on the circle. These rings have maximal ideals $\mathfrak m_p = ...