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

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1answer
49 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
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1answer
51 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
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1answer
39 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
29 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.
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3answers
98 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 ...
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1answer
34 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 ...
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1answer
55 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?
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1answer
78 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 ...
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1answer
48 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?
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1answer
43 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 ...
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2answers
50 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 ...
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1answer
45 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 ...
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1answer
42 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 ...
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1answer
20 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
51 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
48 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
52 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 ...
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1answer
49 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 ...
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1answer
26 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. ...
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1answer
68 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 ...
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1answer
57 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 ...
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1answer
25 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 ...
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1answer
29 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 ...
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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 ...
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1answer
46 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 ...
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1answer
54 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 $0$. ...
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1answer
47 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.
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1answer
25 views

Field of Quotients Explanation

I'm having a hard time grasping the concept of a field of quotients. The book I'm currently reading gives the following definition: Any integral domain D can be enlarged to a field F such that every ...
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1answer
65 views

An integral domain of characteristic $k>0$ is a vector space over $\mathbb{Z}_k$?

Problem statement: Suppose that $R$ is an integral domain of characteristic $k > 0$. Show that $R$ can be considered as a vector space over $\mathbb{Z}_k$. This seems like a trivial ...
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2answers
55 views

Characteristic of Integral-domain where $15a=0$ but $3b\neq 0$.

Let $R$ be an integral domain. Let $a,b \in R$. Assume that a and b both not zeros, $15a = 0$ and $3b \neq 0$ group. What can you say about the characteristic of $R$?
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1answer
29 views

Integral multiplicative system over a domain

Suppose $A$ is a domain and $S\subseteq A$ is a multiplicative system. Show that $S\subseteq A^\times$ if and only if $S^{-1}A$ is integral over $A$. I've started $\Leftarrow$ below... Suppose ...
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1answer
155 views

Is there a characterization of integral domains in terms of the homomorphisms out of them?

In the $\mathbf{Set}$-concrete category of commutative rings, we can define that an object $A$ is a field iff for every homomorphism $f : A \rightarrow B$, precisely one of the following holds. $f$ ...
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1answer
30 views

Show $I=p\mathbb{Z}$ for prime $p$.

Let $I\subset\mathbb{Z}$ be an ideal such that $I\neq \mathbb{Z}$ and if $I\subset J\subset\mathbb{Z}$ then $I=J$ or $J=\mathbb{Z}$. Show that $I=p\mathbb{Z}$ for some prime $p$. Attempt: We know ...
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1answer
72 views

Do the ring of smooth functions on $\Bbb R$ form an integral domain?

Do the ring of smooth functions on the set of real numbers $\Bbb R$ with the usual pointwise addition and multiplication form an integral domain? I have been trying to prove this result without ...
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0answers
31 views

Let $A=\Bbb Q[x,y]/(x^2+y^2−1)$. How to show that Frac$(A)$ is isomorphic to $\Bbb Q(t)$? [duplicate]

Let $A=\Bbb Q[x,y]/(x^2+y^2−1)$ and note that $A$ is a domain. How to show that Frac$(A)$ (i.e. the "field of fractions") is isomorphic to $\Bbb Q(t)$? I know that the parametrization of the ...
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2answers
87 views

$R$ domain is an injective $R$- module $ \Rightarrow $ $R$ is a field [closed]

This is an exercise from Rotman: Prove that if $R$ is a domain and an injective $R$-module, then $R$ is a field. Any hint ?
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1answer
62 views

Abstract Algebra: integral domain and principal ideal domain

I am studying by myself and I needed help for few question which I am confused how give proof of that. Let $\varphi : J \to K$ be a ring epimorphism with $\varphi(1) = 1$, where $J$ and $K$ are ...
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0answers
11 views

Domain GCD Property

Let D be a domain and $\emptyset \subset A \subseteq D^*$ $d \in GCD(A)$ if and only if (d) is a minimum among the principal ideals containing (A) If $d \in GCD(A)$ then d|a for all $a \in A$ and ...
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1answer
22 views

Property of GCD in ring

Let D be a domain and $\emptyset \subset A \subseteq D^*$ Show that CD(A)={$d\in D$ | $(A)\subseteq (d)$} I know that I'll need to show both containments to show that the two statements are ...
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1answer
13 views

GCD property of Domain

Let D be a domain and $\emptyset \subset A \subseteq D^*$ If $x \in D^*$ and $GCD(xA)\neq \emptyset$ then $GCD(A)\neq\emptyset$ and $GCD(xA) = xGCD(A)$. I've already figured out how to show that ...
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0answers
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Properties induced by surjective ring homomorphism between two integral domains

Let $f: R \rightarrow S$ be surjective ring homomorphism between two integral domains. (a) If $R$ satisfies Ascending Chain Condition on Principal Ideals, must $S$ also satisfies Ascending Chain ...
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2answers
111 views

Every submodule of free $R$-module is free; is $R$ an integral domain?

Let $R$ be a commutative ring with identity such that every submodule of every free $R$-module is free. As part of an exercise, I'm trying to show that $R$ is an integral domain. Aiming for a ...
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2answers
71 views

Irreducible elements and Associates

Show that, in a domain, every associate of an atom is an atom. An atom is the same thing as an irreducible element. I think these two facts will be important to prove this statement: A nonunit ...
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2answers
65 views

Why $\Bbb Z$ = (integers) is an integral domain?

Is there a proof for $\Bbb Z$ being an integral domain? Or is it an axiom that if $a*b=0$ then $a=0$ or $b=0$ where $a$, $b$ belong to $\Bbb Z$. Is it so obvious?
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33 views

The domain of a function [closed]

Find the domain of $$ f(x) = \frac{1}{⌊x − 2⌋ + ⌊x − 10⌋ − 8}, $$ where $⌊x⌋$ is the greatest integer less than or equal to $x$. x can't equal to 10?
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1answer
98 views

What is the normalization of the ring $\mathbb C[x,y,t]/(t^3-x^3y)$?

I would like to compute the normalization of the ring $A=\mathbb C[x,y,t]/(t^3-x^3y)$, but I do not know how to proceed. I am not an expert in normalizations, and the only examples I saw were ...
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2answers
80 views

In a noetherian integral domain every non invertible element is a product of ireducible elements

I want to prove that in a noetherian ring $R$ which is also an integral domain, every non invertible element can be expressed as product of ireducible elements. I really do not know where to ...
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0answers
153 views

“M is reflexive” implies “M is maximal Cohen-Macaulay”. Is the converse true?

Let $R$ be a local integrally closed domain of dimension $2$. Let $M$ be a nonzero finitely generated $R$-module. We know that "$M$ is reflexive" implies "$M$ is maximal Cohen-Macaulay". Is the ...
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1answer
52 views

Irreducibility of a polynomial in $F[x,y]$.

This is a rather easy question, but I'm not entirely confident in why I think this is true. Let $F$ be a field. Consider $F[x,y]$. I want to show that the ring $F[x,y]/(y^{2} - x)$ is an integral ...
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1answer
101 views

Isomorphism of field of fractions

Assume that $1\in R$ I have a question which is as follows: Let $R$ be a commutative integral domain and $I$ a non-zero ideal of $R$. Let $F$ be the field of fractions of $R$. Suppose that ...