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

learn more… | top users | synonyms

1
vote
1answer
48 views

If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is $R$ finite?

If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is it true that $R$ is finite ? (I know that there are infinite domains with unity, ...
2
votes
3answers
53 views

$D$ be a UFD, if an element of $D$ is not a square in $D$ then is it true that, that element is not a square in the fraction field of $D$?

Let $D$ be a UFD, let $F$ be the field of fractions of $D$, let $a \in D$ be such that $x^2 \ne a, \forall x \in D$. Then is it true that $x^2\ne a ,\forall x \in F$ ? (This problem is motivated ...
-1
votes
2answers
37 views

Polynomial ring, ideals and Spec

Morning everyone, I want some hint about this. i) Determine all ideals of $\frac{\Bbb{R[X]}}{<X^3-1>}$ where $R$ is real set ii)Is $\frac{R[X]}{<X^3-1>}$ integral Domain iii)...
3
votes
1answer
65 views

Irreducible elements for a commutative ring that is not an integral domain

Why does the definition of an irreducible element require us to be in an integral domain? Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral ...
3
votes
0answers
36 views

Integral closure and $\bigcap \mathfrak{a}^n$

Let $R$ be a domain such that $\bigcap_{n=1}^\infty \mathfrak{a}^n=0$ holds for all proper ideals $\mathfrak{a}$ of $R$ (this holds, for example, if $R$ is Noetherian). Let $K$ be the quotient field ...
1
vote
1answer
41 views

Meaning of $Z\oplus Z$

I am a beginner in Ring Theory and just started Integral Domains. In my textbook, the following was stated : $Z\oplus Z$ is not an integral domain. I can't understand this. I know $\oplus$ ...
6
votes
0answers
142 views

Can we characterize all infinite Euclidean-domains having exactly one invertible element?

$\mathbb Z_2$ and $\mathbb Z_2[x]$ are two euclidean-domains having exactly one invertible element ; my question is ; Can we characterize all euclidean domains $D$ having exactly one invertible ...
-1
votes
2answers
98 views

Prove that $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ is an integral domain [closed]

Let $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ be an analytic algebra. I am trying to prove that $R$ is an integral domain. Basically I know that if $\langle x^2 - yz\rangle$ is a prime ...
0
votes
1answer
87 views

If $R$ is an integral domain and $R[x]$ is an euclidean domain, then $R$ is a field [closed]

Is this obvious? I cannot see that this is true. The converse is fairly obvious though. I tried to show $(x)$ is a maximal ideal and try the quotient but failed. I will appreciate any help.
0
votes
0answers
36 views

Local integral domain which is not regular

This exercise is from Sharp, Steps in Commutative Algebra. Thanks to any hints and answers 15.35. Exercise Let $A=\Bbb R[X_1,\ldots,X_{n+1}]$ (where $n\in\Bbb N$), the ring of polynomials over ...
1
vote
2answers
25 views

The ring of all continuously n-times differentiable functions is not an integral domain for all n?

Let $\mathbb{Z^+}$ be the set of all non-negative integers. For each $n\in \mathbb{Z^+}$ let $S_{n} =C^{n}([0,1], \mathbb{R})$ be the ring of all continuously n-times differentiable functions where $...
4
votes
1answer
145 views

Algebraically Closed Quotient Fields

It is well-known that if the quotient field of a commutative noetherian integrally closed domain $R$ is algebraically closed, then $R$ is a field. The proof is easy: let $r_0 \in R$ and choose $r_i ...
1
vote
0answers
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 $1$...
0
votes
1answer
44 views

Relationship between modules and maximal ideals of a commutative ring [closed]

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 $m=0$?
0
votes
3answers
48 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 -...
-2
votes
1answer
44 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?
0
votes
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 ...
0
votes
2answers
37 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 $AB=(a)$. ...
1
vote
0answers
19 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: $A = \{q,w,e\}$ ...
0
votes
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 ...
1
vote
0answers
32 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 ...
2
votes
0answers
36 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 ...
1
vote
1answer
40 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 $q,...
0
votes
2answers
40 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 $A$-...
0
votes
1answer
50 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 to ...
0
votes
0answers
32 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
64 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
49 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
17 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
44 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
votes
4answers
97 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 $$\mathbb{R}[x,y]/I$$ is ...
6
votes
2answers
177 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), $p(...
0
votes
2answers
51 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 $(V^{-1}...
1
vote
1answer
98 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 ...
17
votes
0answers
200 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
32 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
54 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
30 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 of ...
0
votes
2answers
43 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
29 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 $a=a_{n}x^{n}+\cdots+a_{1}x+a_{0}\in R[x]$ ...
-1
votes
2answers
44 views
0
votes
0answers
89 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
87 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 ...
1
vote
1answer
126 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
72 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 s\...
1
vote
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.
1
vote
2answers
78 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
71 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 is $...
3
votes
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 ...