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

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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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28 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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626 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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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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25 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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138 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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66 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
58 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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94 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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65 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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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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101 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
45 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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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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117 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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231 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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211 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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110 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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140 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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110 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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364 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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2answers
377 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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697 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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149 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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158 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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51 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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476 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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80 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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111 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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314 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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779 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
178 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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171 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 ...
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235 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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184 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 = ...
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179 views

If two elements in a ED have the same Euclidean norm, they are associates?

Is it very obvious that on a Euclidean Domain, two elements $x$ and $y$ have the same Euclidean norm $\nu(x) = \nu(y)$ then they are associates? Can someone give me a proof of this?
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Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain.

Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain. We haven't really discussed principal ideal domains. I've heard that this is easy, but I just ...
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96 views

abstract algebra question concerning Euclidean domains

Let $R$ be a Euclidean domain, and let $r_{1}$, $r_{2}$, $r_{3}, \ldots,r_{n}$ be (distinct) elements of $R$. Prove that there are elements $a_{1}$, $a_{2}$, $a_{3},\ldots,a_{n}$ such that $d = ...
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918 views

A subring of the field of fractions of a PID is a PID as well.

Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID? Any ...
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228 views

Rings such that $A[x]$ is a principal ideal domain

Let $A$ be a commutative ring. Then the following assertions are equivalent. $A$ is a field; $A[x]$ is a Euclidean domain; $A[x]$ is a principal ideal domain; $A[x]$ is a unique factorization ...
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Show $\mathbb{Z}[\sqrt{2}]$ is a PID

I understand how to show something isn't a PID (namely by constructing a counterexample), and I think I understand the proof that $\mathbb{Z}$ is a PID, but I'm not sure how to modify it so that I can ...
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1answer
127 views

Why does the result hold for PIDs but not for UFDs?

Let $R$ be a subring of an integral domain $S$, and suppose $R$ is a PID. Then it follows that if $r\in R$ is a gcd of $r_1$ and $r_2$ in $R$, where $r_1$ and $r_2$ are not both zero, then $r$ is a ...
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2answers
1k views

Greatest common divisor in the Gaussian Integers

Let $a$ and $b$ be integers. Prove that their greatest common divisor in the ring of integers is the as their greatest common divisor in the ring of Gaussian Integers. Ring of Gaussian Integers is: ...
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2k views

Show that every ideal of the ring $\mathbb Z$ is principal

Let $\mathbb Z$ be the ring of integers. The question asks to show that every ideal of $\mathbb Z$ is principal. I beg someone to help me because it is a new concept to me.
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136 views

Queries on proof that every PID is a factorisation domain

I'm reading a proof from C. Musili's Rings and Modules that every PID is a factorisation domain. The author defines a factorisation domain as a commutative integral domain $R$ with a unit such that ...
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358 views

Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
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769 views

Dedekind domain with a finite number of prime ideals is principal

I am reading a proof of this result that uses the Chinese Remainder Theorem on (the finite number of) prime ideals $P_i$. In order to apply CRT we should assume that the prime ideals are coprime, i.e. ...
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486 views

Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

I read this proof that if $D$ is an integral domain and $D[X]$ is a principal ideal domain, then $D$ is a field. My question is if the requirements can be relaxed a bit, namely: Is it true that ...
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183 views

Number of ideals of a PID modulo an ideal

Let $R$ be a Principal Ideal Domain and $(a)\neq(0)$ an ideal of $R$. Prove $R/(a)$ has a finite number of ideals.