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

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2
votes
0answers
32 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 ...
0
votes
1answer
36 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 ...
0
votes
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
votes
1answer
47 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 ...
0
votes
0answers
30 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 ...
0
votes
2answers
34 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
votes
1answer
40 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 ...
1
vote
2answers
10 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
votes
1answer
39 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 ...
-1
votes
0answers
28 views

$(A')_P = (A_P)'$ where $A$ is a finitely generated fractional ideal.

Let $R$ be an integral domain with fraction field $K$, $A$ be a fractional ideal of $R$, and $A'=\{x\in K:xA\subseteq R\}$. If $A$ is a finitely generated $R$-module, prove $(A')_P = (A_P)'$ ...
5
votes
4answers
93 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 $$\mathbb{R}[x,y]/I$$ ...
6
votes
2answers
170 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), ...
0
votes
2answers
46 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 ...
3
votes
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 ...
1
vote
1answer
43 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 ...
16
votes
0answers
159 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
votes
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
votes
0answers
49 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
votes
1answer
27 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 ...
0
votes
2answers
38 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 ...
1
vote
2answers
26 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 ...
-1
votes
2answers
42 views
0
votes
0answers
40 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
votes
1answer
82 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
votes
0answers
28 views

Suppose $R$ is commutative integral domain. If $M$ is a vector space over $Frac(R)$, then it is divisible and torsion free as an $R$-module.

My question concerns part (1) of the exercise below (Exercise 3.15(p. 54) in "Rings and Categories of Modules" F.W. Anderson and K.R Fuller.) I have included the text, and my attempted solution ...
0
votes
1answer
53 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
votes
2answers
65 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 ...
1
vote
0answers
51 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.
1
vote
2answers
46 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
votes
1answer
57 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
votes
3answers
41 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
160 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 ...
0
votes
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
votes
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 ...
-2
votes
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.
-4
votes
1answer
111 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
vote
1answer
52 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
votes
1answer
30 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
votes
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 ...
1
vote
2answers
53 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
96 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
votes
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: ...
1
vote
1answer
33 views

Jacobian of the Transformation Problem, Multivariable Calculus

I have the following Jacobian problem: I'm having trouble working through it because the double integral in terms of u and v is throwing me off. Could someone walk me through it? Thanks!
0
votes
1answer
57 views

Finite integral domain

I encountered a problem: Every finite integral domain is isomorphic to $ \mathbb{ Z }_{p} $. I know that finite integral domain is isomorphic to a field, but I have no idea on how to construct ...
2
votes
2answers
50 views

Integrate $\int\arccos(\sqrt{\frac{x-4}{x+6}})dx$

I need integrate: $$\int\arccos(\sqrt{\frac{x-4}{x+6}})dx$$ How can i solve it? is it good way substitute argument of arccos? $$t=\sqrt{\frac{x-4}{x+6}}$$
0
votes
0answers
34 views

$f \in A[X]\backslash \{0\}$ can be written as $f=af_0$, where $f_0 \in A[X]$ is primitive

Let $A$ be an integral domain. Show that any $f \in A[X]\backslash \{0\}$ can be written as $f=af_0$, where $f_0 \in A[X]$ is primitive. I know how to prove for the case $A = \mathbb{Z}$ be ...
1
vote
0answers
15 views

Is this the correct domain of integration for this double integral, under the following coordinate transformation?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy} \ dx \ dy$, where $A$ is the region defined by $x>0, \ y>0$ satisfying $x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x} ...
0
votes
0answers
23 views

What would the limits of integration be for this double integral?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy}dx \ dy$, where $A$ is the area defined by $x>0, \ y>0, \ x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x^{2}} \leq y \leq ...
2
votes
1answer
78 views

Non-existence of an irreducible fraction representation

If $R$ is a unique factorization domain and $F$ its field of fractions then any $z\in F$ can be represented as an irreducible fraction (i.e. $z=a/b$ with $a,b\in R$ so that $a$ and $b$ have no common ...