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

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27 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 ...
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75 views

Overrings of holomorphy rings

Let $F$ be a function field and $S$ be an arbitrary (and non trivial) subset of the set of places of $F$. Let $H=\bigcap_{P\in S} O_P$, where $O_P$ is the valuation ring associated to the place $P$. ...
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2answers
28 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 ?
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3answers
47 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 ...
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1answer
38 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 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 ...
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1answer
33 views

Looking for an example of a GCD domain which is not 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 ...
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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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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 ...
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1answer
34 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
39 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 ...
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1answer
46 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 ? ...
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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
33 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 ...
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1answer
44 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 ...
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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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16 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 ...
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1answer
45 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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35 views

Show that the defined module is not finitely generated, but can be finitely generated with a modification

Define $R[U^{-1}]=\{\frac{r}{u}|r\in R, u\in U\}$, which is an R-module, where $R$ is an integral domain. In my example if $U=\{s^i|i\in\mathbb{Z}_{>0}\}$ for some $s\in R$, then prove that ...
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3answers
54 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 ...
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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?
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224 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
55 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
28 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 ...
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0answers
52 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 ...
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1answer
31 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 ...
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1answer
64 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
56 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
38 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 ...
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2answers
58 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?
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1answer
184 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!!
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3answers
92 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 ...
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0answers
65 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
21 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, ...
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1answer
98 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 ...
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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] = ...
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1answer
24 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 ...
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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 ...
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1answer
50 views

Finite Extension of Integral Domains.

Let $D\subset E$ (integral domains), with fraction fields $k\subset K $. Suppose that $E$ is integral over $D$, and $E$ is $D$-module finitely generated. My question is: $[K:k]$ is finite? Thank ...
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2answers
329 views

Intuition for why finite integral domain is a field

So I am not getting the intuition for why every finite integral domain is a field. I mean I saw the proof but still I feel like its somehow not intuitive to me of why finiteness of integral domain ...
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1answer
93 views

Does any (noetherian) integral domain have a “UFD closure”?

Let $R$ be a (possibly noetherian if that helps) commutative unital integral domain. Does there exist a UFD $\overline{R}$ such that $R$ embeds in $\overline{R}$ (via some map $\psi$) and such that ...
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1answer
32 views

Showing certaing Integral domain is not well ordered.

Let $\mathbb{Z}[\sqrt2]=\{ a+b\sqrt2 ~| a,b\in \mathbb{Z}\} $ be an integral domain. Let $p=\{ a+b\sqrt2 ~|a,b\in\mathbb{Z} ~~~ and ~~~ a+b\sqrt 2 ~~is~~a~positive~real~number \} $ Show that ...
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1answer
20 views

Show that $\mathbb{Z} [\sqrt p]$ is an ordered Integral Domain.

Let $\mathbb{Z}[\sqrt p]=\{ a+b\sqrt p ~| a,b\in \mathbb{Z},p~is~prime\} $ Assume $\mathbb{Z}[\sqrt p]$ ia an integral domain with usual addition and multiplication. Show $\mathbb{Z}[\sqrt p]$ is an ...
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1answer
58 views

Show that it is an integral domain

I want to show that $$\mathbb{C}[X,Y,Z]/(Y-X^2,Z-X^3) \text{ is an integral domain }.$$ How can I do this? Do I have to find a homomorphism from $\mathbb{C}[X,Y,Z]/(Y-X^2,Z-X^3)$ to an integral ...
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1answer
51 views

Is $K[x,y,z,\frac{1}{xz}]$ an integral domain?

Let $K$ be an Algebraic closed field with not necessarily characteristic 0. I'm interested in knowing if $K[x,y,z,\frac{1}{xz}]$ is an integral domain or not. At first I rewrite my ring as ...
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1answer
43 views

No sub-integral domain of Z with prime characteristic?

I try to find a subring of Z which it is integral domain and characteristic is a prime. Until now, I can't find it. But i believe that this proposition is true. Please help me prove or disprove.
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1answer
30 views

A doubt on a proposition involving Goldman domains.

$(*)$ Let $S/R$ be an extension of domains. Assume that for some $a\in R$, the ring $R[a]$ is Goldman. Then I want to show that $a$ is algebraic over $R$, whence $R$ is also a Goldman domain. DEF A ...
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1answer
99 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
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1answer
24 views

What exactly does it mean for a maximal ideal to be unique in a principal ideal domain?

I'm currently reading about PIDs and have come across a question involving maximal ideals which at one point reads "Suppose that a Euclidean domain $R$ had a unique maxima ideal $P$". Does this mean ...
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2answers
89 views

Prove the fractional field of an integral domain is the smallest field containing the integral domain

I have two questions about the fractional field of an integral domain. Given an integral domain $D$: Is there a difference between saying "the fractional field of $D$ is the smallest field ...