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

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31 views

An example of a c.i./Gorenstein/C.M. integral domain which is not integrally closed

If I am not wrong, it is known that: {Regular rings} $\subsetneq$ {Complete intersection rings} $\subsetneq$ {Gorenstein rings} $\subsetneq$ {Cohen-Macaulay rings}. It is known that a regular ring ...
4
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1answer
65 views

Principal ideal domains that are not integral domains

In the usual definition, a principal ideal domain $R$ is also assumed to be an integral domain? However, the property that every ideal is generated by a single element does not seem to immediately ...
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0answers
58 views

Quotient field - base change

For my master thesis, I need to examine the following statement: $Frac(R) \otimes_{k} L \cong Frac(R \otimes_{k} L)$, where $R$ is an integral domain over the perfect field $k$ and $L$ is a finite ...
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1answer
42 views

Unique isomorphisms and universal properties

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ...
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2answers
36 views

Does $I(J\cap K)=IJ\cap IK$ hold in a finitely generated polynomial $K$-algebra for $K$ a field?

Let $K$ be a field and $R:=K[X_1,X_2,\cdots, X_n]$ for a certain $n\in\mathbb N$. If $I,J,K$ are three ideals of $R$, can we conclude that $I(J\cap K)=IJ\cap IK$?
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2answers
21 views

Let $R $ be an integral domain with identity. Prove that if $p $ is irreducible and $u$ is a unit, then $pu $ is irreducible.

Let $R $ be an integral domain with identity. Prove that if $p $ is irreducible and $u$ is a unit, then $pu $ is irreducible. My proof: Let $ Pu=nm$ $P=u^{-1} nm$ Then $u^{-1} n$ or m is unit ...
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2answers
113 views

Does $IJ=IK\implies J=K$ always hold for integral domain and finitely generated nonzero ideal $I$?

Let $R$ be a commutative integral domain, $I,J,K$ three ideals of $R$ with $I\neq (0)$ being finitely generated. Then does $IJ=IK$ imply $J=K$? With Nakayama lemma, I can prove it if one of $J$ and ...
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0answers
26 views

Proving that for an integral domain $R$, $y\in (x)\iff (y)\subseteq (x)$.

I am trying to prove the following statement. Let $R$ be a integral domain. Then for all $x,y\in R$ we have $$x\mid y\iff y\in(x)\iff (y)\subseteq (x).$$ Note that $(x)$ denotes the principal ...
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1answer
40 views

Show that $R$ is an integral domain iff for all $ x, y, z\in R, xy = xz $ implies $y = z$.

Suppose $R$ is a commutative ring. Show that $R$ is an integral domain iff for all $ x, y, z\in R, xy = xz $ implies $y = z$. Proof: $\Rightarrow $Let $x,y,z\in R$ such that $x(y-z)=0$ ...
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1answer
18 views

Congruences and the integers as an integral domain

I am trying to prove the following: If $p$ is prime, then the only solutions of the congruence $x^2 \equiv x$ (mod $p$) are those integers $x$ such that $x \equiv 0$ (mod $p$) or $x \equiv 1$ (mod ...
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2answers
55 views

Does $\text{End}_R(I)=R$ always hold when $R$ is an integrally closed domain?

Let $R$ be a commutative ring with identity, and $I\neq 0$ an ideal of $R$, I'm thinking how to calculate $\text{End}_R(I)$. I have proved that when $R$ is a integral domain, ...
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2answers
104 views

Noetherian ring under some conditions has at least two minimal prime ideals

Question is : Suppose $R$ is a noetherian ring. Prove that $R$ is either an integral domain, has nonzero nilpotent elements, or has at least two minimal prime ideals. [Use the previous exercise.] ...
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2answers
131 views

Basic question about finite fields and characteristic

I am reading Herstein's "Topics in Algebra" and I've encountered with the following problem: If $D$ is an integral domain and $D$ is of finite characteristic, prove that the characteristic of $D$ is ...
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1answer
55 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 ...
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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
72 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 ...
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2answers
93 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 ...
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1answer
45 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 ...
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2answers
61 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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45 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
42 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 ...
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1answer
37 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 ...
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1answer
31 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 ...
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1answer
33 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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2answers
54 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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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
55 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 ...
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1answer
59 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 ...
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1answer
105 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
23 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
48 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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42 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
55 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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38 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
47 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
54 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
27 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
66 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
27 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
50 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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59 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
25 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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249 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
59 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
35 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
58 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
37 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 ...