For questions regarding integral domains, their structures, and properties.

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8 views

$\gcd({p_1}^{n_1},{p_2}^{n_2})$ associates with $1$ if $p_1,p_2$ are prime and do not associate

$R$ is a integral domain, $p_1$ and $p_2$ are prime, $p_1$ and $p_2$ do not associate, $n_1,n_2 \ge 1,n_1,n_2 \in \mathbb N $, I need to show that $g:=\gcd({p_1}^{n_1},{p_2}^{n_2})$ associates with ...
0
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1answer
37 views

Relationship between modules and maximal ideals of a commutative ring

Let $A$ be an integral domain, $M$ an $A$-module, and $m\in M$. Now for all maximal ideals $\mathfrak{m}$ there exists an $n\notin \mathfrak{m}$ such that $nm=0$. Why does this mean that $m=0$?
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3answers
42 views

How do I show that the integral domain $Z/(p)$ is not an ordered integral domain?

The properties of an ordered integral domain are the following: 1. Closed under addition in $D^+$ 2. Closed under multiplication $D^+$ 3. Satisfy the trichotomy law which either $a=0,a\in D^+, or ...
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1answer
41 views

Can infinite integral domain be a field [closed]

Every finite integral domain is field. I was wondering can some infinite integral domain be field?
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0answers
20 views

Unit elements of polynomial ring [duplicate]

If D is an integral domain. What are unit elements in polynomial ring $D[x]$? Are unit elements of $D[x]$ equal to unit elements of $D$? And what if D is a field? Then unit element should be nonzero ...
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2answers
34 views

Cancellation law of ideals in a certain ring

Let $R$ be a integral domain satisfying the following property. For any non-zero ideal $A$ of $R$, there exist $a \in R\ (a \neq 0)$ and a non-zero ideal $B$ of $R$ such that $AB=(a)$. ...
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0answers
17 views

Find Range of Domain and Co-Domain

Good Afternoon, I was wondering if someone could explain to me how to find the range of a domain and co-domain? For example, if A represents my domain and B represents my co-domain: $A = \{q,w,e\}$ ...
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0answers
22 views

Are there any integral domains in which irreducible elements are easily identified?

In every integral domain I've studied so far irreducible elements have been impossible to quickly identify in general with any known procedure. Is there an integral domain for which such a procedure ...
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0answers
31 views

Let a,b have the same divisor (content) in an integral domain A. When can I deduce $a/b\in A^\times$?

Given a Noetherian integral domain A and a finitely generated torsion A-module M, we can define the divisor, or content, of M to be $div(M)= \sum_{P, ht(P)=1} \ell(M_P) [P]$, where the sum ranges ...
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0answers
33 views

What is the intuition behind a Euclidean function?

Many algebra textbooks give the definition of a Euclidean domain as an integral domain $R$ equipped with a Euclidean function/map (let's call it $\nu$). What I don't understand is the significance of ...
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1answer
39 views

Prove or disprove that the ring $\mathbb{Z}[x]/(x^2-1)$ is an integral domain

Prove or disprove that the ring $\mathbb{Z}[x]/(x^2-1)$ is an integral domain. It is easy to see that $\mathbb{Z}[x]/(f)=\{P+(f): \deg(P)<\deg(f)\}$. If $\deg(P) \geq \deg(f)$, there exists ...
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2answers
38 views

Extension of the fields of fractions of integral domains

Let $A$ and $B$ be integral domains, and let $\varphi:A\to B$ a ring homomorphism. We can give $B$ a structure of $A$-module by saying $ab = \varphi(a)b$. Suppose that $B$ is a finitely generated ...
0
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1answer
48 views

Determine the integral closure of a ring.

Let $R=F[X,Y]/(Y^2-X^3)$. Determine the integral closure of $R$ in its quotient field. I guess I should reduce the problem to some statement related to $F[X]$. For $F$ of characteristic not equal ...
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0answers
31 views

Definition of an Integral Domain in the second edition of Herstein's Topics in Algebra

I think a definition in Herstein's Topics in Algebra needs to be modified and I am asking this question to make sure I am not missing something. Herstein defines an integral domain as a commutative ...
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2answers
51 views

Let $D$ be a principal ideal domain. Show that every proper ideal of $D$ is contained in a maximal ideal of $D$.

I know that a PID must satisfy the Ascending chain condition. So Im guessing its going to involve that in the argument some way but Im not sure how to prove it.
2
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1answer
43 views

Example of finite field extension where root not separable

My class notes has as theorem (without proof): "Let $K/F$ be finite field extension, with $K=F(\alpha_1,\ldots,\alpha_n)$ and $\alpha_k$ is separable for all $k$. Then $K/F$ is separable". My ...
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2answers
11 views

Existence of GCD in UFD

I proved that any two elements in PID have GCD and it can be expressed as linear combination of those two elements. I know that even in case of UFD GCD exists but it may not be expressed as linear ...
0
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1answer
40 views

In an Integral Domain is it true that $\gcd(ac,ab) = a\gcd(c,b)$?

In my algebra class I was given as homework assignment to prove that: Given an integral domain $A$ and $a,b,c,d,e \in A$ then if $d = \gcd(b,c)$ and $e = \gcd(ac, ab)$ then $e = ad$. It is easy ...
5
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4answers
94 views

Show $\mathbb {R}[x,y]/(y^2-x, y-x)$ is not an integral domain

Let $\mathbb{R}[x,y]$ denote the polynomial ring in two variables $x$, $y$ over $\mathbb{R}$, and let $I = (y^2-x,y-x)$ be the ideal generated by $y^2-x$ and $y=x$. Show that $$\mathbb{R}[x,y]/I$$ ...
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2answers
173 views

Which are integral domains? Fields?

Which of the following rings are integral domains? Which ones are fields? (a) $\mathbb{Z}[x]/(x^2 + 2x +3)$ (b) $\mathbb{F}_5[x]/(x^2+x+1)$ (c) $\mathbb{R}[x]/(x^4+2x^3 +x^2 +5x+2)$ For (a), ...
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2answers
48 views

The polynomial of minimal degree with root $\alpha$ is unique.

So I am working on the following proof: Problem Statement: Let $\alpha$ be a complex number. Prove that the kernel of the substitution map $\mathbb{Z}[x] \rightarrow \mathbb{C}$ that sends $x ...
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1answer
53 views

Are rings of fractions of integral domains closed under finite intersection?

Let $D$ be an integral domain with fraction field $K$. Let $V$, $W$ be multiplicatively closed subsets of $D$. Consider the rings of fractions $V^{-1}D$ and $W^{-1}D$ as subrings of $K$. Is ...
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1answer
54 views

Prove that the Gaussian Integers are an integral domain

We have the following Theorem: A non-zero commutative ring is an integral domain if and only if for all $a$,$b$ $\neq 0$ $\implies ab \neq 0$. Now, we need to prove that the Gaussian integers form an ...
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0answers
174 views

Where Fermat's last theorem fails

It's fairly well known that Fermat's last theorem fails in $\mathbb{Z}/p\mathbb{Z}$. Schur discovered this while he was trying to prove the conjecture on $\mathbb{N}$, and the proof is an application ...
0
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1answer
30 views

Order of an element in an integral domain

Suppose that (R, + , •), is an integral domain, where + and • are the usual operations, addition and multiplication respectively, and the non zero element r, as considered as an element of the abelian ...
0
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0answers
50 views

Prime element and irreducible

I would like to know whether a polynomial in $\mathbb Z[x]$ is a prime element if and only if it is irreducible. Since $\mathbb Z[x]$ is an integral domain, a prime element in $\mathbb Z[x]$ is ...
0
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1answer
29 views

The type automorphisms over $D[X]$, where $D$ is an integral domain. [closed]

If $D$ is an integral domain, then show that every automorphism $f$ of $D[X]$ which is identity on $D$ is of the form $f(X)=cX+d$, where $c$ is a unit of $D$. It is easy to show that a function ...
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2answers
40 views

Prime ideal and maximal ideal of an integral domain

In an integral domain, $\{0\}$ is always a prime ideal. What about maximal ideal? $(0)$ is a prime ideal in $\mathbb{Z}$ which is an I.D but it is not maximal in $\mathbb{Z}$. So I can not conclude ...
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2answers
27 views

Identifying units in a polynomial ring

Problem Statement: Let $R$ be a domain. Identify the units in $R[x]$. I am trying to identify the units in a domain $R$ by considering an arbitrary element $a=a_{n}x^{n}+\cdots+a_{1}x+a_{0}\in ...
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2answers
42 views
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0answers
69 views

Krull dimension of finitely generated algebra over field

Let $k$ is a field and $A = k[x_1, x_2, ..., x_n]$ is finitely generated $k$ algebra, which is also an integral domain. Then I know that $A$ has finite Krull dimension (equal to the transcendence ...
3
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1answer
86 views

If any two nonzero elements of an integral domain R are associates, prove that R is a field?

Question from abstract algebra course...I have already shown that if a=bu for some unit u and b=av for some unit v, by definition of associates, then u and v are inverses...but I can't quite make the ...
0
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1answer
78 views

How to show every field is a Euclidean Domain.

I'm having trouble proving this. This is what I have so far: Let $F$ be a field. Let $v(x) \rightarrow 1$ for all $x$ not equal to $0$. So if we let $x$ be in $F$ where $x$ not zero then we can ...
2
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2answers
68 views

Definition of gcd's in Euclidean domains

In a course, we defined $\gcd(a,b)$ in a Euclidean domain to be a common divisor of $a,b$ with greatest possible norm/valuation. Looking at a (commutative) ring $R$ as a category with $r\rightarrow ...
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0answers
52 views

Give an example of a homomorphism from an integral domain $R$ to a field which does not factor through the inclusion $R\to\mathrm{Frac}(R)$.

All homomorphisms from fields to fields factor through this inclusion. Thus, the integral domain must not be finite. This is all I have gotten so far. Best regards.
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2answers
61 views

use the definition for an integral domain to prove that Z7 is an integral domain

Integral Domain Divisors 7 is prime 〖[a]〗_7 〖[b]〗_7=〖[0]〗_7∈Z_7 〖[ab]〗_7=〖[0]〗_7 ab∈[0]_7 ab is multiple of 7 a∈[0]_7 and b ∈ 〖[0]〗_7 ...
2
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1answer
61 views

$R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor , then is $R$ an integral domain?

Let $R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor (i.e. if $P$ is a prime ideal and $x,y \in P$ with $xy=0$ then either $x=0$ , or $y=0$). Then ...
3
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3answers
46 views

$R$ be an infinite commutative ring with unity such that for every non-zero ideal $I$ , $R/I$ is finite ; then is $R$ a PID or at least Noetherian?

Let $R$ be an infinite commutative ring with unity such that for every non-zero ideal $I$ of $R$ , $R/I$ is finite; then is $R$ a PID or at least Noetherian ? I can only prove that $R$ must be an ...
4
votes
3answers
161 views

nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
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2answers
24 views

Show that the set of all principal ideals is an equivalence class of the relation $\sim$

Let $A$ a integral domain and let $\mho(A)$ the set of all non-zero ideals. Show that the set of all principal ideals is an equivalence class of the relation $\sim$ that we can noted by $[A]$. ...
0
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0answers
17 views

Can an iterated integral over a box R ={(x,y,z)|x∈[0,a], y∈[0,b], z∈[0,c]} be expressed in eight different ways?

this is my first time on stack exchange so sorry if I am not following any guidelines. I received this exact question on a midterm and answered yes, it is possible, which was considered wrong on the ...
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1answer
102 views

Commutative domain with two maximal ideals of different heights [closed]

Give an example of a commutative domain $R$ and two maximal ideals $\mathfrak{m}_1, \mathfrak{m}_2$ in $R$ of different heights.
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1answer
133 views

Every nonzero element $x\in\mathbb Z_n$ is either a unit or a zero divisor [closed]

I'd like to show that every nonzero element $x\in\mathbb Z_n$ is either a unit or a zero divisor, i.e. for every $x\in\mathbb Z_n$ there exists either $x'\in\mathbb Z_n$ such that $x'x=1$, or ...
1
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1answer
53 views

Prove whether R is integral domain.

I'm having trouble with figuring out whether a given ring is an integral domain or not. This comes from my confusion about the zero element. This ring R is a commutative triple $(Z,*,o)$ with ...
0
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1answer
31 views

the associate of a prime is prime in integral domain

I was hoping someone could give me a hand getting started trying to prove that in an integral domain, if a and b are associates, then a is prime if and only if b is.
5
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1answer
53 views

$A\subseteq B$ integral domains with surjective multiplication, then the localization by all monic polynomials evaluated at some point is nonzero

Let $A\subseteq B$ be two integral domains such that the multiplication function $m: A \times (B \setminus A) \to B$, $m(x,y)=xy$, is surjective. Let $S \subset A[x]$ be the set of all monic ...
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2answers
60 views

Prove that in any GCD domain every irreducible element is prime

The proof of the following proposition is not completely clear to me. I get everything up until the bold part and I have a feeling some crucial steps are omitted, can anybody help clear this up? ...
4
votes
2answers
97 views

Invertible ideals are finitely generated.

Let $R$ be an integral domain and let $I,J \subseteq R$ be ideals. Suppose $IJ=(a)$ for some $a \in R$. We wish to show that $I$ and $J$ are finitely generated. Since $a \in IJ$ we know $a$ can ...
2
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2answers
67 views

Proving $\mathbb C[x,y]/\langle x^2+y^2+1\rangle,\mathbb R[x,y]/\langle x^2+y^2+1\rangle$ are integral domains

As a homework assignment, I need to prove that $\mathbb C[x,y]/\langle x^2+y^2+1\rangle$ and $\mathbb R[x,y]/\langle x^2+y^2+1\rangle$ are integral domains. I have no idea how to approach ...
0
votes
1answer
28 views

find domain of $\int\arccos \left(\sqrt{\frac{x-4}{x+6}}\right)$

How can i find domain of this integral? $$\int\arccos\left(\sqrt{\frac{x-4}{x+6}}\right)dx$$ I tried this: $$\frac{x-4}{x+6}\ge0\implies (-\infty,-6)\cup [4,\infty)$$ Next: ...