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

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About $R/I$ Where $I$ is a Prime Ideal

A well known result in Commutative Algebra says: for a commutative ring $R$ with $1$, $R/I$ is an Integral Domain if and only if $I$ is a Prime Ideal of $R$. Can this result be generalised for non ...
3
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1answer
125 views

Can an element in a Noetherian domain have arbitrarily long factorizations?

I tried to answer this question two days ago. Unfortunately, I said ring, rather than domain, which is what I wanted. So I try again. Let $R$ be a Noetherian commutative domain and let $r\in R$. ...
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3answers
21 views

Transformation to polar coordinates in an integral

Suppose that the domain of integration for a double integral is: $\{(x,y), - \infty < x \le a, -\infty < y \le a \}$. If I want to do a change of variable (to polar coordinates), how do I ...
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3answers
123 views

Rings and Fields

I have a few questions in ring and fields theory. First of all, I was trying to show that the field of quotients of $\frac{\mathbb{Z}_{12}}{\langle 4 \rangle}$ is exactly itself, once it is a field. ...
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0answers
19 views

What is the multiplicative order of 1+sqrt(2) in Z[sqrt(2)]? [duplicate]

I want to know that 1+sqrt(2) in Z[sqrt(2)], I am not sure what is multiplicative order.please guide also multiplicative order also. Actually I am in context of Contemporary Algebra by Joseph A ...
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2answers
80 views

What is the multiplicative order of $1+\sqrt{2}$? [closed]

Actually I am in the context of Contemporary Algebra by Gallian, where there is topic of divisibility in integral domains, where there is inverse of $1+\sqrt{2}$ in $\mathbb Z[\sqrt{2}]$. I understand ...
2
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1answer
16 views

Does every ID with subring which has a unity have an unity?

For an arbitrary ID (integral domain) $R$ with subring $S$, assume that $S$ has an unity. Then does $R$ have a unity too? If not, please provide a counter-example.
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1answer
33 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
71 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
59 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
44 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
39 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 ...
3
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2answers
116 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
27 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
45 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
20 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 ...
2
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2answers
56 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
107 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
132 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 ...
2
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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 ...
3
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3answers
54 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
76 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
97 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 ...
3
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2answers
62 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
24 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
46 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
46 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
40 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
37 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
60 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
74 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
56 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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3answers
83 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 ...
3
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1answer
107 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
49 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
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
60 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
41 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
50 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
59 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
71 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
28 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
52 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 $f=g \iff f(x)= g(x), \forall x \in R$. $\bf Attempt:$ ...