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Questions tagged [integral-domain]

For questions regarding integral domains, their structures, and properties. This tag should probably be accompanied by the Ring Theory tag. This tag is not for use for questions regarding integrals in analysis and calculus.

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How Should I show that these $k$-algebras are not Isomorphic?

Question Show that the $k$-algebras $k[x,y]/\langle xy \rangle$ and $k[x,y]/\langle xy-1 \rangle$ are not isomorphic. Attempt At first, I thought $xy=0$. This would mean both $x$ and $y$ are zero ...
Mr Prof's user avatar
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3 answers
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$\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$?

I am exploring whether the following assertion holds true in integral domains: $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$. Let us make this formal below. Consider two elements $a$ and $b$ in an ...
Martin Geller's user avatar
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$R = \mathbb R[X,Y]/(XY - 1)$ and $I$ be the ideal of $R$ generated by the image of the element $X - Y$ in $R$. Describe $R/I$

Let $R = \mathbb R[X,Y]/(XY - 1)$ ($\mathbb R$ is the set of real numbers) and I be the ideal of R generated by the image of the element X - Y in R. I want to find a way to describe R/I, i.e. find a ...
Jishnu's user avatar
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Proof verification: Is the quotient ring $\Bbb C[x]/(x^2+1)$ an integral domain? [duplicate]

Is the quotient ring $\Bbb C[x]/(x^2+1)$ an integral domain? My solution goes like this: If possible let us assume that $\Bbb C[x]/(x^2+1)$ an integral domain. This means $(x^2+1)$ is a prime ideal in ...
Thomas Finley's user avatar
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Show that if $R$ is an integral domain and $f(x)$ is a unit in the polynomial ring $R[x]$, then $f(x) \in R$ [duplicate]

Show that if $R$ is an integral domain and $f(x)$ is a unit in the polynomial ring $R[x]$, then $f(x) \in R$. Proceed by contraposition. Suppose $R$ is an integral domain, $f(x)$ is a unit in $R[x]$, ...
Grigor Hakobyan's user avatar
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Regarding Vakil's Exercise $10.7.A$

$\newcommand{\O}{\mathscr{O}}\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\Frac}{\operatorname{Frac}}$ In exercise 10.7.A of Vakil's Rising Sea we are tasked with showing that if $A$ is an ...
Chris's user avatar
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3 votes
1 answer
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Simplifying Gauss’s Lemma

Hello Math StackExchange Community, I am revisiting Gauss's Lemma in my lecture notes and considering a simplification in its proof. I am proposing to remove the necessity of proving that $\lambda$ is ...
Martin Geller's user avatar
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60 views

$A\otimes_kB$ is an integral domain when $k$ is algebraically closed, and $A$ and $B$ are finitely generated

I am trying to understand this answer in this math stack exchange post I don't see how it works. Let $A$ and $B$ be finitely generated $k$ algebras with $k$ algebraically closed, and suppose that $A$ ...
Chris's user avatar
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Noether normalization and free modules

Let $A$ be a finitely generated algebra over some field $k$, and assume that $A$ is an integral domain. By Noether normalization, we have a finite injective ring homomorphism $\varphi: k[x_1, \ldots, ...
Adelhart's user avatar
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Rings that appear as quotients $B/I$ of subrings$B \subseteq F$ of fields $F$ and for $I \subseteq B$ ideals

What are the rings $A$ that appear as quotients $B/I$ of subrings$B \subseteq F$ of fields $F$ and for $I \subseteq B$ ideals? For each $A$, give an explicit formula for a ring $B$ and a field $F$. ...
love and light's user avatar
3 votes
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If $R$ for a commutative ring with identity (not necessarily an integral domain) satisfies ACCP, then does $R[x]$ also have ACCP?

I read here that if $R$ is an integral domain and has ACCP, then $R[x]$ also has ACCP. However, is this necessarily true for a commutative ring with identity? If it is false, then what is a ...
852619's user avatar
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Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?

Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
Alex's user avatar
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Existence of prime elements in an atomic integral domain

Let $R$ be an integral domain, is it true that if $R$ is atomic, then it must contain a prime element? If not, what is a counterexample? I know that if an element is prime, then if $I$ is the ideal ...
852619's user avatar
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definition of characteristic of an integral domain

Consider the following definitions taken from the book Topics in Algebra by I. N. Herstein. An integral domain $D$ is said to be of characteristic $0$ if the relation $ma=0$, where $a\neq 0$ is in $D$...
MathArt's user avatar
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factorization into coprimes subordinate to two given coprimes in GCD domain

Let $a,b,c$ be elements in a GCD domain (or just a GCD monoid) $R$. Suppose that $\text{gcd}(a,b)=1$ and $c \ne 0$. If $R$ is a UFD, it is possible to write $c=a'b'$ with $\text{gcd}(a',a)=\text{gcd}(...
Junyan Xu's user avatar
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Counterexample of Divisibility of Ideals with Product of Ideals

Given a commutative ring $R$ with unity, we define for $I,J\subseteq R$ ideals $I\ \vert\ J\iff I\supseteq J$ $IJ=\{\sum_i a_ib_i:a_i\in I, b_i\in J\}$ For every commutative unitary ring $R$ it ...
tripaloski's user avatar
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Is this the correct negation of the definition and criterias for Irreducible elements in integral domain?

Background: Definition: If $R$ is an integral domain, $p\in R$ is called an irreducible element (and is said to be irreducible in $R$) if it satisfies the following conditions: (1) $p\neq 0$ and $p$ ...
Seth's user avatar
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1 vote
1 answer
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ID's, PID's, Noetherian rings and valuation rings: implications amongst them

I am trying to establish some implications between being an ID, a PID, a Noetherian ring and a valuation ring. First of all, I know that PID $\Rightarrow$ Noetherian, because in a PID every ideal is ...
kubo's user avatar
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What is the meaning of the abbreviation id(x,y)

In the subject of the change of variables via integration of 2-forms on $\mathbb{R}^2$ for a function $\omega=f(x,y) dx \wedge dy$ over $D \subseteq \mathbb{R}^2$, the teacher says that we go from a ...
M. Lemelin's user avatar
3 votes
2 answers
118 views

Is kernel of evaluation map on $R[x]$ always principal if $R$ is an integral domain?

Suppose $R$ is an integral domain and $E \supseteq R$ is a ring extension of $R$. Let $\alpha \in E$ commute with all of $R$, and consider the evaluation homomorphism $\phi_\alpha : R[x] \to E$ given ...
MathNeophyte's user avatar
3 votes
1 answer
87 views

Prove that if $P$ is a prime ideal of a non-commutative von Neumann regular ring $R$, then $P$ is a maximal ideal

Let $R$ be a ring with identity elements (not necessarily commutative). A ring $R$ is called a von Neumann regular ring if for every $a\in R$ there is $b\in R$ such that $a=aba$. How do I prove that ...
Belajar Matematika's user avatar
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If $R$ is a ring such that $\forall x\in R, x^n=x$ for some $n>1$ then when $P$ is prime, why is $R/P$ finite?

I was recently reading a post on MSE which had an argument like: If $P$ is a prime ideal of a ring $R$ all of whose elements satisfy $x^n = x$, then $R/P$ is an integral domain with the same property....
Thomas Finley's user avatar
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Prove $\cap_{n\geq0} I^n = (0)$

Let $A$ be an integral domain and noetherian and let $I \subset A$ a proper ideal such $I*\cap_{n\geq0} I^n = \cap_{n\geq0} I^n$. Prove that $\cap_{n\geq0} I^n = (0)$ I'm trying to prove it by getting ...
Juan José Campos's user avatar
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Free submodules of an integral $R.$

I was told by the author of the answer here Showing that the rank of $M$ is exactly $1.$ that: Free submodules of an integral domain $R$ are exactly the principal ideals of $R.$ I am wondering which ...
user avatar
3 votes
1 answer
287 views

Every finite integral domain is a field (why is it commutative?)

Okay, a finite integral domain is a finite ring $D$ such that for every $a, b \in D, ab = 0$ iff $a = 0$ or $b = 0$. We want that $D$ is a field, meaning it has a unit element ($1$), every element is ...
Chuwee's user avatar
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1 answer
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Number of zero divisors of a ring [duplicate]

I am given the ring $\mathbb{Z}_8 \times \mathbb{Z}_{10}$, and I am asked to find the number of zero divisors. I have counted $23$, not manually, but taking into account several aspects such as what ...
Emmy N.'s user avatar
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-1 votes
2 answers
185 views

Prove that a ring of order $6$ can never be an integral domain. [duplicate]

Prove that a ring of order $6$ can never be an integral domain. My solution: Let $R$ be a ring of order $6$ which is an integral domain. This means, that $1+1\neq 0\in R$ and we note that, $(1+1)(1+1+...
Thomas Finley's user avatar
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0 answers
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In $f(x)= (x-a_1)(x-a_2)(x-a_3)$, why are the only roots $a_1,a_2,$ and $a_3$ over a prime modulus? [duplicate]

I am trying to justify why if we have a polynomial $f(x)= (x-a_1)(x-a_2)(x-a_3)$ over a prime modulus $p$,it's only roots are $a_1, a_2,$ and $a_3$. Why are there no other values of $x$ which could ...
Princess Mia's user avatar
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1 vote
2 answers
247 views

Error in Herstein's "Topics in Algebra", zero ring has characteristic $1$ which is not prime

I was trying to prove the following statement: "If an integral domain has a finite characteristic then the characteristic of the integral domain is a prime number" This made me look at the ...
Thomas Finley's user avatar
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0 answers
60 views

Non-toric ring which is Cohen-Macaulay

I know a lot of examples of classes of binomial ideals $I$ in $S=K[x_1,\dots,x_n]$ whose $S/I$ is a Cohen-Macaulay domain. Basically, if $I$ is a toric ideal and there exists a monomial order $<$ ...
Hola's user avatar
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0 answers
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Does there exist a surjective but not injective homomorphism of rings from an integral domain R → R

I could come up with examples for rings;say for example defining the homomorphism to be a left shift of elements which are taken to be of the form of a countably infinite tuple.Can someone help me out ...
Bavanesh B S's user avatar
5 votes
0 answers
127 views

When does a semiring extend to an integral domain?

Mirroring the construction of $\mathbb{Z}$ from $\mathbb{N}$, we can extend a commutative and additively cancellative semiring $A$ to its additive group of differences, $B$, and then define ...
Alex's user avatar
  • 913
1 vote
2 answers
43 views

gcd(a S)= a gcd(S) (integral domain or ufd) [duplicate]

Les $A$ be an integral domain[1] and $S\subset A$. The set $gcd(S)$ is the set of elements $\delta$ satisfying the two following properties : $\delta|s$ for all $s\in S$ if $d|s$ for all $s\in S$, ...
Laurent Claessens's user avatar
4 votes
1 answer
265 views

Integral domains with isomorphic field of fractions

Suppose I have two integral domains $A, B$ which have isomorphic fields of fractions: $$F := \text{Frac}(A) \cong \text{Frac}(B).$$ We have canonical injections $$\iota_A: A \hookrightarrow F$$ $$\...
Rivers McForge's user avatar
2 votes
0 answers
42 views

"Completion" of an integral domain with irreducible nonprime elements

I was trying to get a feel for what it means for an element to be irreducible but not prime, and I came across the idea that an irreducible nonprime element $r$ has "hidden" factors. That is,...
eyeballfrog's user avatar
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0 votes
1 answer
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The difference between two classes of a function

I have a function f. What is the difference between the below cases: a$\in$ $C^0$[0,1]$\cap$$C^1(0,1]$ a$\in$ $C^1$[0,1] I know that $C^0$[0,1] means that the function is continuous. But what is the ...
i.issa's user avatar
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1 vote
0 answers
53 views

Is $I^n \not = I^{n+1}$ for all non-zero proper ideals $I$ of an integral domain? [duplicate]

For $R$ a (commutative with 1) integral domain, is it possible to have $I^n = I^{n+1}$ for some non-zero proper ideal $R$? I realise that in the case of a Noetherian domain, we can apply Krull's to ...
George's user avatar
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2 votes
1 answer
230 views

Show that $K[x_1,x_2,x_3,x_4] / \langle x_1x_4 - x_2x_3 \rangle$ is an integral domain of dimension $3$

I am stuck at the following exercise from Gathmann's notes on Algebraic Geometry on page 21: Let $R = K[x_1,x_2,x_3,x_4] / \langle x_1x_4 - x_2x_3 \rangle$. Show that $R$ is an integral domain of ...
3nondatur's user avatar
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0 votes
1 answer
222 views

Tensor product of a polynomial ring and the quotient of a polynomial ring modulo a prime ideal

Let $S,R$ be two polynomial rings over a field $K$ in a finite number of variables, and let $I$ be a prime ideal of $R$. Then is it true that the tensor product over $K$ between $S$ and $R/I$ is a ...
Hola's user avatar
  • 185
3 votes
1 answer
131 views

$A[X, Y ]/(X^2+Y^2−1)$ is integral domain

Deduce that if $A$ is a unique factorization domain (UFD) of char- acteristic zero, then $A[X, Y ]/(X^2+Y^2−1)$ is an integral domain. Let $B = A[X, Y]/(X^2 + Y^2 - 1)$, and suppose that $f(X, Y)$ ...
Abcd's user avatar
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1 vote
1 answer
94 views

Is an element of a quotient field of $\mathbb{Z}$ a number or a set?

In my book, it is mentioned that the Quotient Field of the Integral Domain of Integers is the Field of Rational Numbers. However, I have a confusion- the Quotient Field of $\mathbb{Z}$ would be the ...
Pourush Sood's user avatar
2 votes
2 answers
190 views

Cancellative property in integral domains

I am currently studying integral domains and see that if $ab = ac$ and $a \neq 0$, then we have $b = c$. I was wondering whether the same is true for three nonzero subsets $I, J, K$ of an integral ...
Ratanjit 's user avatar
3 votes
1 answer
379 views

How does partial fraction expansion generalize to fractions of integers? Why is it not unique, in that case?

The Wikipedia page for partial fraction expansion mentions that it can be generalised to "regular" fractions, i.e. fractions of integers: https://en.wikipedia.org/wiki/...
Theo H's user avatar
  • 257
4 votes
1 answer
462 views

Is there the terminal object in the category of integral domains?

I wondered what the initial object and the terminal object are in the category of integral domains. Simple argument: Since integral domains do not put an additional restriction on the definition of a ...
Dannyu NDos's user avatar
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0 votes
0 answers
217 views

Definition of the kernel of a ring homomorphism with 1

I came across this problem: Let $R$ be a integral domain. Show that the map $b \mapsto ba$, with $a \neq 0$, is injective. My proof went as follows: Since $R$ is an integral domain, we have $ab = 0 \...
Kajice's user avatar
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1 vote
1 answer
62 views

Find domain of function whose independent variable is the upper limit of an integral?

I'm trying to answer the following question: Find the domain of $F(x)$ where $$F(x)=\int_{5}^{x}\frac{1}{1-t^2}dt$$ Are we supposed to use some type of convergence test? I've heard of that for series, ...
user3925803's user avatar
0 votes
1 answer
37 views

Finitely generated subalgebras of an algebraic closure [closed]

Let $k$ be a field and $\bar{k}/k$ be an algebraic closure of $k$. Let $k'$ be a finitely generated $k$-algebra, which is a subalgebra of $\bar{k}$. Is it true that $k'$ is a finite field extension of ...
MarkInderes's user avatar
1 vote
0 answers
50 views

Tensor Product of a polynomial quotient field with itself

Let $\mathbb{Q}$ be the rational numbers, and let $k = \mathbb{Q}[X]/(X^3-2)$. Describe the $\mathbb{Q}$-algebra $k \otimes_{\mathbb{Q}} k$ as explicitly as possible. Is it a domain? We have been told ...
Utkarsh Gupta's user avatar
0 votes
1 answer
98 views

A finite integral domain with more than one element is a field (Proof explanation)

Here is the question I want to answer: Let $R$ be a finite commutative ring with more than one element and no zero-divisors. Show that $R$ is a field. I do not understand some ideas in the proof given ...
Emptymind's user avatar
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1 vote
1 answer
53 views

Is this my drawing of domain of $\{(x,y)\in\mathbb{R^2}\mid-{1\over 2}\leq y\leq x\leq{1\over 2}\}$ correct?

I want you to check my depiction of the region shown below of multiple integral is correct or not. I think that it seems correct but cannot have a strong confidence. $$ A:=\left\{(x,y)\in\mathbb{R^2}~\...
electrical apprentice's user avatar

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