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

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0
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1answer
28 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
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1answer
20 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
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0answers
61 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 ...
1
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1answer
42 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 ...
5
votes
2answers
282 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 ...
4
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1answer
66 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 ...
1
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1answer
21 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 ...
0
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1answer
17 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 ...
2
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1answer
53 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 ...
1
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1answer
43 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 ...
-1
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1answer
38 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.
1
vote
1answer
26 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 ...
1
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1answer
84 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 ...
1
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1answer
20 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 ...
1
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2answers
61 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 ...
1
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1answer
27 views

how to define size function in Euclidean domain

I was reading about examples of Euclidean domains and their proofs. I encountered one problem on how to define size function for various Euclidean domains. For example in $\mathbb{Z}[i]$ size ...
1
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1answer
48 views

Does $ax\in\mathfrak{m}I$ with $x\in I\setminus\mathfrak{m}I$ and $a \in R$ imply $a\in\mathfrak{m}$ for an invertible fractional $R$-ideal $I$?

Let $R$ be an integral domain, $\mathfrak{m}$ a maximal ideal of $R$, and $I$ an invertible fractional $R$-ideal. If $x \in I \setminus \mathfrak{m}I$ and $a \not\in \mathfrak{m}$, do we have $ax ...
7
votes
1answer
78 views

Generalizing the Big Omega function to Integral Domains

The $\Omega(n)$ function counts the total number of prime factors of $n$ counting multiplicity. Obviously, this definition extends to any Unique Factorization Domain. I have two follow up questions: ...
0
votes
1answer
55 views

Unique factorization into irreducible elements?

Given is $R$ an integral domain. With $P$ we will denote the subset of the irreducible elements of $R$, such that for every irreducible $x\in R$ exactly one element of $\left \{ ux\mid u\in ...
0
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1answer
35 views

In the integral domain $D = \{r+s \sqrt{17}: r,s \in\Bbb Z\}$, which element is irreducible?

In the integral domain $D = \{r + s \sqrt{17} | r,s \in\Bbb Z\}$, which is irreducible? $3 - \sqrt{17}$ $9 - 2\sqrt{17}$ $7 + \sqrt{17}$ $13 + \sqrt{17}$ I got all of them are irreducible, if you ...
1
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1answer
49 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
2
votes
1answer
28 views

Domain and double integral

Let $$D = \{(x,y)\in R^2 : 0<x<y<2x,x^2+y^2>4,xy<4\}$$ and $f : D \rightarrow R$ the continus and bounded function defined by $f(x,y)=xy$ I'm stucked to find some bounds for $\iint_D ...
0
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0answers
15 views

Morandi's Rings appendix: about a step of the proof that that $R[x]$ is a UFD if $R$ is. [duplicate]

In the appendix about rings in Patrick Morandi's book Fields and Galois Theory, we find the following exercise (which arises in the proof of the theorem: $R[x]$ is a UFD if $R$ is a UFD). Let B be ...
1
vote
1answer
55 views

About units in $\mathbb{Z}[\sqrt[3]{2}]$

Is it true that all units in $\mathbb{Z}[\sqrt[3]{2}]$ are of form $\pm(1+\sqrt[3]{2}+\sqrt[3]{4})^n$ for some integer $n$ ?
2
votes
1answer
59 views

Ring structure on $\mathbb{Z}$

Is there a possibilty to define a (nontrivial) ring structure on $\mathbb{Z}$ (or $\mathbb{Q}, \mathbb{R}$) other than the usual so that $\mathbb{Z}$ (or $\mathbb{Q}, \mathbb{R}$) with that structure ...
2
votes
1answer
71 views

Let $F$ be a field and let $R$ be the $F$-subalgebra of $F[x]$ generated by $x^2$ and $x^3.$ Show that $R$ is not a unique factorization domain

$1.$ Let $F$ be a field and let $R$ be the integral domain in $F[x]$ generated by $x^2$ and $x^3.$ Show that $R$ is not a unique factorization domain Attempt: $R = \langle x^3,x^2 \rangle = ...
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1answer
32 views

Equivalence of a integer domain

If $R$ is a commutative ring with $1$. Suppose that for all polynomial $P(X)\in{R[X]\setminus{R}}$ has at most $n$ roots, with $n=grad\ (f)$ then $R$ is an integer domain. Any suggestion, please.
3
votes
3answers
103 views

Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$?

Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then ...
3
votes
1answer
37 views

If an integral domain $R$ has a factorization basis, is it a UFD?

By a factorization basis for an integral domain $R$, let us mean a subset $\xi$ of the commutative monoid $R^\times = R \setminus \{0\}$ such that firstly, no two elements of $\xi$ are associates, and ...
-3
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1answer
66 views

One dimensional noetherian domain

Let $(R,m)$ be a one-dimensional Noetherian domain. Is $R$ a regular or a topical ring like Gorenstein or other kinds?
2
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1answer
106 views

Exercise from Kaplansky's Commutative Rings and Eakin-Nagata Theorem

Exercise 15 of section 2-1 of Kaplansky's Commutative Rings is to show that if $T$ is a Noetherian ring and is finitely generated module over a subring $R$ of $T$, then $R$ is Noetherian. Kaplansky ...
0
votes
1answer
59 views

If the localization of a ring is a field, then the ring is an integral domain?

Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?
2
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1answer
47 views

For some finitely many nonzero prime ideals, the contraction and extension of their product is zero

I was reading P.M. Eakin's thesis paper, The converse to a well known theorem on Noetherian Rings. The following is taken from Theorem 2, page 281 of that paper, and that's where I'm stuck. Let ...
4
votes
2answers
84 views

the nonzero elements of Z3[i] form an abelian group of order 8 under multiplication. Is it isomorphic to Z8??

$\mathbb{Z}/3\mathbb{Z}[i]$ is an integral domain, so its characteristic is a prime number. But, in order to prove that it is isomorphic to $\mathbb{Z}_8$, we have to show that $\mathbb{Z}_3[i]$ has ...
1
vote
1answer
48 views

Definition of irreducible element.

In my abstract algebra book, it says that an element $p \in R$ where $R$ is a commutative ring is irreducible if $\textbf{(i)}$: $p$ is not a unit $\textbf{(ii)}$: if $p=ab$ then either $a$ or $b$ is ...
2
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1answer
59 views

Facts about quotient rings - example

I have three quotient rings: $R_1 = \frac{\mathbb{Q}[x]}{(x^2 -1)}$ $R_2 = \frac{\mathbb{Q}[x]}{(x^2 +1)}$ $R_3 = \frac{\mathbb{Q}[x]}{((x -1)^2)}$ I am trying to decide whether these are integral ...
2
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1answer
21 views

Show that if $R$ is principal, $N $ is pure and $Ann(x+N)= Rd$ then there exists $y \in M$ such that $x+N=y+N$ and $Ann(y)=Rd$

Let $R$ a integral domain and $M$ a $R$-module. A submodule $N$ of $M$ is pure if for all $x \in M$ and $a \in R$ such that $ax \in N$, there exists $y \in N$ such that $ax=ay$. Show that if $R$ ...
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2answers
67 views

Show that an integral domain $R$ is principal if and only if every submodule of a cyclic $R$-module is cyclic.

Good morning, I have difficulty with this problem: Show that an integral domain $R$ is principal if and only if every submodule a cyclic $R$-module is also cyclic.
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2answers
56 views

Integral domain (rings and fields)

Let $\mathbb Z[i]=\{a+ib \mid a, b \in \mathbb Z \}$. How to Show that $\mathbb Z[i]$ is a integral domain?
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1answer
66 views

Height of finitely generated ideals in a catenary local ring

If $R$ is a noetherian local domain which is catenary, and $a_1,...,a_n$ are elements of the maximal ideal of $R$ with $\operatorname{height}(a_1,...,a_n)=n$, could we conclude that ...
0
votes
1answer
50 views

Dimension of a certain vector space

I have an integral domain $R$ with field of fractions $K$. Let $M$ be a finitely generated free module over $R.$ If $M$ has an $R$-basis $\{u_i\}_{i=1}^d$ then is it true that the set $\{1\otimes_R ...
0
votes
1answer
30 views

Prove that if $\mathrm D$ is integral domain, then $\mathrm D$is UFD iff these conditions hold.

Prove that id $\mathrm D$ is integral domain, the "$\mathrm D$ is UFD " iff (1) $\mathrm D$ satisfies the ACC for principal ideals. (2) every irreducible element is a prime element. ...
1
vote
1answer
85 views

$2$-dimensional Noetherian integrally closed domains are Cohen-Macaulay

Any 1-dimensional Noetherian domain is Cohen-Macaulay (C-M). For the $2$-dimensional case, a condition of being integrally closed is necessary to be added for a Noetherian domain to be C-M, which ...
1
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1answer
66 views

A question on localization of fractional ideals

I have a domain $A$ with field of fractions $K$ and a non-zero fractional $A$-ideal $I$. Let $I'$ be the fractional ideal $\{a\in K\mid aI\subset A\}$. I assume that $II'\subset \mathfrak p$ for ...
3
votes
1answer
29 views

Are these properties for a monoid enough for being the underlying monoid of an integral domain minus the zero?

If $R$ is an integral domain then $R-\{0\}$ equipped with the original multiplication can be recognized as a commutative and cancellative monoid. The inversible elements form a subgroup $R^*$ and it ...
1
vote
1answer
31 views

Showing that if $F$ an integral domain, $F[x_1,\dots,x_n]$ is an integral domain

If $F$ is an integral domain, show that $F[x_1,\dots,x_n]$ is an integral domain. Can anyone help out by giving me hints on how I should show that is an integral domain? I would appreciate ...
0
votes
0answers
31 views

Know a formula for $x \pmod{y + z}$?

It seems natural to want to add the $x \pmod y$ operator to the integers (or other generalization of the integers?), but perhaps it has no algebraic properties. Can you prove that there is no ...
2
votes
1answer
52 views

Question about Euclidean Domain but not a Field

Here's a question that I'm wondering about: Let $A$ be a Euclidean domain with Euclidean function $\delta$, but $A$ is not a field. Is it true that $\delta \left({A^*}\right)$ (where $A^* = A ...
2
votes
1answer
55 views

For $A$ integral domain, $S$ multiplicative subset, if $x/s \in S^{-1}A$ is irreducible, then $x$ is irreducible.

I need to prove the following: For $A$ integral domain, $S$ multiplicative subset, if $x/s \in S^{-1}A$ is irreducible, then $x$ is irreducible. First some observation: $x$ cannot be a unit nor $0$. ...
0
votes
1answer
67 views

Maximal ideal generated by irreducible element

Let $R$ be an integral domain and let $(c)$ be a non-zero maximal ideal in $R$. Prove that $c$ is an irreducible element.