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

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2answers
56 views

Applications of $\mathbb{Z}/n\mathbb{Z}$ [on hold]

I would like someone to proof me this claim and give me its applications in mathematics if it's not a convention. Claim: for all positive integers $n$, the ring $\mathbb{Z}/n\mathbb{Z}$ is domain if ...
2
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1answer
48 views

A question on zero divisors [closed]

In the ring $\mathbb Z_n$, the divisors of zero are precisely those elements $m\in \mathbb Z_n$ such that $(m,n) > 1$. Proof: Let $d = (m,n)$ and note that ...
3
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3answers
53 views

In a commutative ring with identity, if $p$ is irreducible, is ($p$) a maximal ideal?

In a Euclidean Domain, $D$, if we mod out by an irreducible, $p$, we get the field $D/(p)$. I can see that this follows since we are going to be able to write $1$ as a linear combination of $p$ and ...
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2answers
60 views

Proving something it NOT and integral domain

Let $R$ and $S$ be two commutative rings with unity. Prove that $R\times S$ is NOT an integral domain. This is the best I could think of so far, please give me a push in the right direction and ...
2
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2answers
72 views

Extending an automorphism to the integral closure

I need some help to solve the second part of this problem. Also I will appreciate corrections about my solution to the first part. The problem is the following. Let $\sigma$ be an automorphism of ...
1
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1answer
39 views

Minimality of field of fractions expressed by functor

I'm probably just below the needed amount of prominent examples to begin studying category theory, but first of all I can't hold back the intrigue, and second I might even benefit from having "arrow ...
3
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2answers
55 views

Prove or disprove $R= \mathbb Q[x]/\langle x^3-x^2+x-1 \rangle$ is an integral domain.

I've got $R$ is not a field since the polynomial is reducible in $ \mathbb Q[x]$. Is it possible to say anything from this?
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1answer
21 views

Prove $R$ conatins an ideal that is not finitely generated. $R = F[x,x^2 y,\ldots,x^n y^{n-1},\ldots]$

Prove R conatins an ideal that is not finitely generated. $R = F[x,x^{2}y,\ldots,x^n y^{n-1},\ldots]$ and is a subring of $F[x,y]$ where $F$ is a field. Seems like $R$ itself is not finitely ...
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1answer
19 views

$R$ has no zero divisors . Let $a \in R$ , $0 \ne b \in R$ and $0 \ne n \in \mathbb Z$ be such that $na+ab=0$ , then is it true that $a=0$?

Let $R$ be a commutative ring (not necessarily with unity) with no zero divisors . Let $a \in R$ , $0 \ne b \in R$ and $0 \ne n \in \mathbb Z$ be such that $na+ab=0$ , then is it true that $a=0$ ?
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2answers
42 views

Rings isomorphic to $\mathbb{Z}_6\times\mathbb{Z}_{10}$

What are five ring properties that hold for each ring that is isomorphic to $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$, but not for every ring? Suppose $Q\approx R$. Then $Q$ has unity, $Q$ is not a ...
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2answers
40 views

Show that $S$ is a field

I'm trying to prove the following result: Let $R$ be a principal ideal domain, $S$ an integral domain and $f: R\to S$ a surjective morphism. Prove that if $f$ is not an isomorphism, then $S$ is a ...
0
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1answer
32 views

If $P$ is a prime then $R/P$ is an integral domain.

I know the same question has been already asked here. So, I am not asking for any proof rather to find out what's wrong with my proof. So, this is what I did: Let, $a+p, b+p \in R/P$, since $P$ is a ...
2
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1answer
27 views

Integral domains examples

I am supposed to give an example of 1) an infinite integral domain of characteristic $5$, and 2) an integral domain which is not a field. Respectively, examples I chose were $\mathbb{Z}_5$ and ...
1
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1answer
29 views

On non-constant multiplicative norms on integral domain and when does the absolute value of the norm is unity implies the element is unit?

Consider $\mathbb Z[\sqrt {d}]$, where $d$ is any non - square integer, define $$N(a + \sqrt d b) = a^2 - db^2 = (a + \sqrt d b)(a - \sqrt d b)$$ as $\mathbb Z \subseteq \mathbb Z[\sqrt {d}]$, so from ...
3
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2answers
44 views

Is it true that an integral domain $R$ is a UFD if and only if intersection of any two principal ideals of $R$ is principal ?

Is it true that an integral domain $R$ is a UFD if and only if intersection of any two principal ideals of $R$ is principal ?
3
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3answers
57 views

Integral Domains and Unique Factorisation Domains

I'm learning about Rings, commutative rings, IDs, UFDs, etc with each being a subset of the predecessor, and I'm now trying to find an ID that is not a UFD I understand $\mathbb Z[\sqrt{-5}]$ is an ...
2
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1answer
53 views

How to show that $\mathbb Z+x \mathbb Q[x]$ is a GCD domain?

How to show that $\mathbb Z+x \mathbb Q[x]$ is a Bezout domain, that is, the sum of two principal ideals is again a principal ideal ? Or at least, how to show that it is a GCD domain ? (This will then ...
1
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1answer
46 views

Looking for an example of a GCD domain which is not a UFD

I know that every UFD (unique factorization domain) is a GCD domain i.e. g.c.d. of any two elements, not both zero, exists in the domain. I am looking for an example of a GCD domain which is not ...
2
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1answer
94 views

An integral domain with Krull dimension 1 which is neither Noetherian nor integrally closed

It seems like a common exercise to try and find rings which only satisfy some of the conditions in the definition of a Dedekind domain. Rings that satisfy exactly 2 of the three conditions were very ...
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3answers
39 views

Concerning $Frac((Frac \space D)[x])$ and $Frac(D[x])$ for an integral domain $D$

Is the fraction field of $\mathbb Z[x]$ a proper subfield (or isomorphic to a proper subfield) of the fraction field of $\mathbb Q [x]$ ? In general , what can we say about $Frac((Frac \space D)[x])$ ...
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1answer
22 views

Is it true that for any integral domain $D$ , $Frac(D)[x] \cong Frac(D[x])$ ?

Is it true that for any integral domain $D$ , $Frac(D)[x] \cong Frac(D[x])$ ? , where $Frac$ denotes the fraction field of the integral domain . I am at a complete loss , please help . Thanks in ...
4
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1answer
36 views

A noetherian local ring having a height one principal prime is a domain

$A$ is a commutative ring with with $1$. If $A$ is a Noetherian and local ring and $A$ has a principal prime ideal of height $1$ then show that $A$ is a domain. Can anybody give some hint.I tried ...
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2answers
40 views

Let $R$ be a commutative ring with no zero divisors, then $R$ can be embedded in an integral domain $S$.

Let $R$ be a commutative ring with no zero divisors, then $R$ can be embedded in an integral domain $S$. I am facing a problem to find the monomorphism $f: R \to S$. Will the function $f(a) ={a ...
0
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1answer
50 views

Does validity of Bezout identity in integral domain implies the domain is PID ?

Let $D$ be an integral domain such that for any $a,b \in D$ , $Da+Db$ is a principal ideal , then must $D$ necessarily be a principal ideal domain i.e. should all the ideals of $D$ be principal ? ...
3
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0answers
32 views

Extension of Integral Domains

Let $S\subset R$ be an extension of integral domains. If the ideal $(S:R)=\{s\in S\mid sR\subseteq S\}$ is finitely generated, show that $R$ is integral over $S$. My first attempt was to show ...
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3answers
44 views

Quotient ring of complex polynomials and ideal domain

Let f(X) = X^2 − 2X + 5 ∈ C[X] and the ideal generated by f(X) be I = f(X)C[X]. (where C(X) is the set of complex polynomials) Prove that the quotient ring C[X]/I is not an integral domain. Since ...
4
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1answer
50 views

Intersection of ring and prime ideal

Give an example of an extension $B/A$ of rings, with $B$ an integral domain and a nonzero prime ideal $\mathfrak{p}$ of B such that $\mathfrak{p} \cap A=(0).$ I don't know where to begin with this.. ...
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1answer
26 views

Manipulations of Euclidean domains

I am trying to answer the following question For (a) I have said that a and ab are in the ring R, by the definition of a ring. Therefore, by the definition of a Euclidean domain a=abq+r. As we are ...
3
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2answers
60 views

Existence of Zero Divisors in $C(X,\mathbb{R})$

Consider any topological space $X$ and $\mathbb{R}$ be with usual topology. The set of all continuous functions from $X$ to $\mathbb{R}$, denoted by $C(X,\mathbb{R})$, is a commutative ring with unity ...
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0answers
22 views

Domain analysis - on and off points on boundaries

I cannot figure out the following: OFF Point: An OFF point of a boundary lies away from the boundary. However, while choosing an OFF point, we must consider whether a boundary is open or ...
3
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1answer
48 views

In $R[x]$, $f=g \iff f(x)=g(x), \forall x \in R$

Let $R$ be an integral domain and $R[x]$ the polynomial ring over $R$. Let $f,g \in R[x]$ such that $\max(\deg f, \deg g)< \#R$. Show that $f=g \iff f(x)= g(x), \forall x \in R$. $\bf Attempt:$ ...
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3answers
55 views

Prove that if $a$ divides both $2$ and $\sqrt{10}$ in $\mathbb{Z}[\sqrt{10}]$, then $a$ is a unit

Prove that if $a$ divides both $2$ and $\sqrt{10}$ in $\mathbb{Z}[\sqrt{10}]$, then $a$ is a unit. Further, show that you can't express $a$ as $a = 2b + \sqrt{10} c$ where $b, c \in ...
1
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1answer
24 views

Homomorphism from $A[X,Y]$ to $A[X]$ with kernel $(X^i-Y^j)$

Let $A$ be an integral domain, $i,j \in \mathbb N$ such that gcd$(i,j)=1$. How would one define a homomorphism from $A[X,Y]$ to $A[X]$, having the ideal generated by $X^i-Y^j$ as its kernel?
11
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2answers
229 views

Show that $A[X]/(aX+b)$ is an integral domain

Let $A$ be an integral domain, $a$ and $b \in A-\{0\}$, and let $B = A[X]/(aX+b)$. Show that, if $Aa \cap Ab=Aab$, then $B$ is an integral domain. My attempt at proof (following a hint). Denote ...
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0answers
56 views

Algebraic characterization of commutative rings with Krull dimension=1,2, or 3

A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about zero-dimensional rings. I could not ...
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1answer
30 views

Proving that the char(R) is non-zero.

Let $R$ be an integral domain and assume that for some non-zero $a \in R$, that $\exists n_a \in \mathbb{N}$ such that $n_a a = 0$. Prove that $R$ has non-zero characteristic. So here is my thinking ...
3
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0answers
55 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions ...
0
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1answer
34 views

Congruence problem in the Euclidean domain $\Bbb Z[\zeta]$

Let $\zeta = \frac12 +\frac{\sqrt{3}}{2}i$. I've proven that $\Bbb Z[\zeta]$ is a Euclidean domain with the norm given by multiplication by the complex conjugate. I'd now like to solve the system of ...
2
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1answer
77 views

Polynomial ring, prime ideal, factor ring

I want to prove that this ideal: $I=(y^3-xz, xy^2-z^2, x^2-yz)$ is prime in $K[x,y,z]$. I think it would be a good idea to prove that the factor ring $K[x,y,z]/I$ has no zero divisors. In this factor ...
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1answer
58 views

Is it possible to characterize the theory of Integral domains with first-order logic alone ?

Is it possible to characterize general ring theory with first-order logic alone ? Is it possible to do so for the theory of Integral domains ?
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2answers
41 views

Is $\mathbb Z$ the only proper sub-domain ( a subring that is an integral domain ) with unity of the ring $\mathbb Q$?

Is $\mathbb Z$ the only proper sub-domain ( a subring that is an integral domain ) with unity of the ring $\mathbb Q$ ? ( I can easily prove that if $D$ is any subring with unity then $\mathbb Z ...
4
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2answers
66 views

GCD in Gaussian integers.

If you have two different common divisors in an integral domain that is not a multiple of each other, is the gcd then equal to the divisor that has the largest norm?
0
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1answer
185 views

Automorphism of an integral domain extends to an automorphism of the quotient field [closed]

Every automorphism of an integral domain can be extended to an automorphism of its quotient field. Please help to start with the proof!!
1
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3answers
96 views

A question about the relation between division ring and domain

"Is it true that any division ring is a domain?" Note 1: I am not sure "domain"="integer domain", are they different? Note 2: Since the definition of integral domain, I can't see if a division ring ...
2
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0answers
67 views

Finite Ring with unity and no zero divisors is field [duplicate]

I would like to know if someone can help me with this. "Show that a finite ring $R$ with unity $1\neq 0$ and no divisors of 0 is a field." The original exercise asked me to show that it was a ...
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2answers
24 views

$A$ integral domain implies $A[x]$ integral domain (proof check)

I'm studying for my abstract algebra course and want to prove as an exercise that if $A$ is an integral domain then $A[x]$ is an integral domain. I realized later that there is a more direct proof, ...
4
votes
1answer
100 views

Finite integral domains are commutative?

Here, integral domain is a non-zero ring $R$ (not necessarily commutative, and not necessarily contains unity), in which $ab=0$ implies $a=0$ or $b=0$. Question If $R$ is a finite integral domain, is ...
0
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1answer
33 views

Why is [b,a] the multiplicative inverse in the field of fraction of an integral domain?

$F = (D \times D^{\ast})/\sim = \{[a,b] | b \neq 0\}$ where $[a,b]$ denotes the equivalence class of $(a,b)$. Define addition and multiplication as $[a,b] + [c,d] = [ad + bc, bd]$ and $[a,b][c,d] = ...
3
votes
1answer
25 views

$f:R \to D$ a homomorphism of the additive group of rings , $f(aba)=f(a)f(b)f(a) , f(1_R)=1_D$ , then is $f$ a ring homomorphism?

Let $R$ be a ring with multiplicative identity $1$ and $D$ be an integral domain with multiplicative identity ( i.e. $D$ is a commutative unital ring without zero divisors ) , let $f:R \to D$ be a ...
2
votes
0answers
70 views

Localization of euclidean ring is euclidean?

I am trying to prove that a localization of a euclidean ring is euclidean, and the converse statement. I feel the basic definition of the norm is enough but I do not know how. Please note I am very ...