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

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5
votes
3answers
136 views

nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
0
votes
2answers
18 views

Show that the set of all principal ideals is an equivalence class of the relation $\sim$

Let $A$ a integral domain and let $\mho(A)$ the set of all non-zero ideals. Show that the set of all principal ideals is an equivalence class of the relation $\sim$ that we can noted by $[A]$. ...
0
votes
0answers
17 views

Can an iterated integral over a box R ={(x,y,z)|x∈[0,a], y∈[0,b], z∈[0,c]} be expressed in eight different ways?

this is my first time on stack exchange so sorry if I am not following any guidelines. I received this exact question on a midterm and answered yes, it is possible, which was considered wrong on the ...
-4
votes
1answer
79 views

Commutative domain with two maximal ideals of different heights [closed]

Give an example of a commutative domain $R$ and two maximal ideals $\mathfrak{m}_1, \mathfrak{m}_2$ in $R$ of different heights.
-4
votes
1answer
47 views

Every nonzero element $x\in\mathbb Z_n$ is either a unit or a zero divisor [closed]

I'd like to show that every nonzero element $x\in\mathbb Z_n$ is either a unit or a zero divisor, i.e. for every $x\in\mathbb Z_n$ there exists either $x'\in\mathbb Z_n$ such that $x'x=1$, or ...
1
vote
1answer
43 views

Prove whether R is integral domain.

I'm having trouble with figuring out whether a given ring is an integral domain or not. This comes from my confusion about the zero element. This ring R is a commutative triple $(Z,*,o)$ with ...
0
votes
1answer
24 views

the associate of a prime is prime in integral domain

I was hoping someone could give me a hand getting started trying to prove that in an integral domain, if a and b are associates, then a is prime if and only if b is.
5
votes
1answer
47 views

$A\subseteq B$ integral domains with surjective multiplication, then the localization by all monic polynomials evaluated at some point is nonzero

Let $A\subseteq B$ be two integral domains such that the multiplication function $m: A \times (B \setminus A) \to B$, $m(x,y)=xy$, is surjective. Let $S \subset A[x]$ be the set of all monic ...
0
votes
2answers
31 views

Prove that in an integral domain, if every two elements have a gcd, every irreducible element is prime

The proof of the following proposition is not completely clear to me. I get everything up until the bold part and I have a feeling some crucial steps are omitted, can anybody help clear this up? ...
4
votes
2answers
87 views

Invertible ideals are finitely generated.

Let $R$ be an integral domain and let $I,J \subseteq R$ be ideals. Suppose $IJ=(a)$ for some $a \in R$. We wish to show that $I$ and $J$ are finitely generated. Since $a \in IJ$ we know $a$ can ...
1
vote
2answers
51 views

Proving $\mathbb C[x,y]/\langle x^2+y^2+1\rangle,\mathbb R[x,y]/\langle x^2+y^2+1\rangle$ are integral domains

As a homework assignment, I need to prove that $\mathbb C[x,y]/\langle x^2+y^2+1\rangle$ and $\mathbb R[x,y]/\langle x^2+y^2+1\rangle$ are integral domains. I have no idea how to approach ...
0
votes
1answer
26 views

find domain of $\int\arccos \left(\sqrt{\frac{x-4}{x+6}}\right)$

How can i find domain of this integral? $$\int\arccos\left(\sqrt{\frac{x-4}{x+6}}\right)dx$$ I tried this: $$\frac{x-4}{x+6}\ge0\implies (-\infty,-6)\cup [4,\infty)$$ Next: ...
1
vote
1answer
27 views

Jacobian of the Transformation Problem, Multivariable Calculus

I have the following Jacobian problem: I'm having trouble working through it because the double integral in terms of u and v is throwing me off. Could someone walk me through it? Thanks!
0
votes
1answer
48 views

Finite integral domain

I encountered a problem: Every finite integral domain is isomorphic to $ \mathbb{ Z }_{p} $. I know that finite integral domain is isomorphic to a field, but I have no idea on how to construct ...
2
votes
2answers
47 views

Integrate $\int\arccos(\sqrt{\frac{x-4}{x+6}})dx$

I need integrate: $$\int\arccos(\sqrt{\frac{x-4}{x+6}})dx$$ How can i solve it? is it good way substitute argument of arccos? $$t=\sqrt{\frac{x-4}{x+6}}$$
0
votes
0answers
32 views

$f \in A[X]\backslash \{0\}$ can be written as $f=af_0$, where $f_0 \in A[X]$ is primitive

Let $A$ be an integral domain. Show that any $f \in A[X]\backslash \{0\}$ can be written as $f=af_0$, where $f_0 \in A[X]$ is primitive. I know how to prove for the case $A = \mathbb{Z}$ be ...
1
vote
0answers
15 views

Is this the correct domain of integration for this double integral, under the following coordinate transformation?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy} \ dx \ dy$, where $A$ is the region defined by $x>0, \ y>0$ satisfying $x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x} ...
0
votes
0answers
23 views

What would the limits of integration be for this double integral?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy}dx \ dy$, where $A$ is the area defined by $x>0, \ y>0, \ x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x^{2}} \leq y \leq ...
2
votes
1answer
78 views

Non-existence of an irreducible fraction representation

If $R$ is a unique factorization domain and $F$ its field of fractions then any $z\in F$ can be represented as an irreducible fraction (i.e. $z=a/b$ with $a,b\in R$ so that $a$ and $b$ have no common ...
3
votes
2answers
40 views

Maximal and Prime Ideals

I was assigned these problems for homework to designate if they were maximal, prime or neither. I was able to determine that (a) was solely prime by showing $\mathbb{Z}[x] /(x-1)$ is isomorphic to ...
2
votes
1answer
19 views

If $R$ is an integral domain, then $(R[x])^\times=R^\times$

If $R$ is an integral domain, then $(R[x])^\times=R^\times$ So since $R$ is an integral domain, it follows that $R[x]$ is an integral domain. We have $f(x)g(x)=1$ then we know that $\deg(f(x)g(x))= ...
1
vote
1answer
76 views

$a\odot b = ab+a+b$ over $\Bbb Z$ no zero divisors?

Define the operations $\oplus$ and $\odot$ on $\Bbb Z$ by $a\oplus b = a+b-1$ and $a\odot b = ab+a+b$. Prove that $(\Bbb Z,\odot,\oplus)$ is an integral domain. Surely they first want me to prove its ...
1
vote
2answers
24 views

Show that each element of $K$ can be written as $\frac{b}{a}$ where, $b\in B, a\in A$

Let $A \subset B$ be integral domains and $B$ is a finitely generated $A$-module. Let $K$ be the fraction field of $B$. Show that each element of $K$ can be written as $\frac{b}{a}$ where, $b\in B, ...
1
vote
0answers
59 views

Generators of polynomial rings in relation to a quotient ring.

I'm struggling with the following problem. Any help is greatly appreciated! Let $R$ be a local integral domain with the unique maximal ideal $\mathfrak m$. Let $g(x), h(x)$ be monic polynomials in ...
2
votes
2answers
27 views

Prove that dimension of $A$ is zero iff it is a field, where $A$ is an integral domain.

Let $A$ be an integral domain. Show that $\dim(A)=0 \iff A$ is a field. The backward implication is trivial. For the forward implication, if we can show that $1 \in <a>$, where $a(\neq 0) ...
1
vote
1answer
28 views

Relation between a maximal ideal and an invertible ideal

Let $D$ be a domain, $\mathfrak{a,b,p}\subsetneq D$ ideals with $\mathfrak{p}$ maximal and $\mathfrak{a}$ invertible (there is some $\mathfrak{c}$ ideal with $\mathfrak{ac}=\mu D$, with $\mu\in ...
0
votes
1answer
23 views

Show the set of GCD's of two elements in an integral domain is the set of associates of their GCD

Sorry if this seem trivial, but im slightly stuck: Let $a$ and $b$ be elements of an integral domain "$R$" and let $d$ be the $\gcd(a,b)$. Show that the set of GCD's of $a$ and $b$ is the set of ...
1
vote
1answer
49 views

Derivative with respect to a variable that is in the region of integration

I need to calculate $f(k)=\frac{dg(k)}{dk}$, where $g(k)=\iiint_{\{(x,y,z)\in\mathbb{R}^3:h(x,y,z)\leq k\}}e^{-(x+y+z)}dxdydz$ $h(x,y,z)\leq k$ is the domain of integration (and quite nasty, by the ...
1
vote
1answer
41 views

A principal maximal ideal

Let $(R,m)$ be a local integral domain, and $t\in m^{-1}$ be such that $tm=R$. Is it true that $m$ is principal? If $t=a/b$ with $a,b\in R$ and $b\not =0$ then $ac/b=1$ for some $c\in m$, and ...
3
votes
2answers
84 views

Showing that if $(a,b)=1$ and if $a\mid c$ and $b\mid c$ then $ab \mid c$, in GCD domains

Is there a proof for the problem below? $R$ is a commutative, integral domain with unity in which for each pair $a,b\in R$, g.c.d. $(a,b)$ exists. I want to show that if $(a,b)=1$ and if $a\mid c$ ...
0
votes
2answers
55 views

General Question about Integral domains.

How do you prove that a particular set is an integral domain? is it enough to prove that there are no zero divisors to say that it is not an integral domain? For example: a + bsqrt(2) : a,b are ...
0
votes
1answer
42 views

Is D/A an integral domain?

Let D be an integral domain and let A be a proper ideal of D. Is D/A an integral domain? If yes, prove it; otherwise, give a counterexample. What I did is: Let (x+A),(y+A) be in D/A where x,y are in ...
2
votes
1answer
47 views

If $V,W$ are valuation domains, then $V+W$ is a ring.

I am stuck at solving the following exercize: Let $V,W$ be two valuation domains with the same fraction field $Q$. Then $V+W$ is a subring of $Q$ (hence it is the smallest subring of $Q$ ...
2
votes
2answers
27 views

is $\Bbb{Q}$ a GCD Domain?

Wikipedia says every UFD is a GCD domain. As every field is a UFD so must be GCD domain. Therefore in rationals, gcd(2,7) must exist. But what is it? I can divide both of them by any rational number. ...
8
votes
1answer
147 views

How can we show that all the solutions are given by $X_m(a), Y_m(a)$?

Let $F$ be an integral domain with characteristic $2$. Let $a\in F[t]$ and $a \notin F$. Let $\alpha (a)$ be a root of the equation $x^2+ax+1=0$. We define two sequences $X_m(a), Y_m(a) \in F[t], m ...
1
vote
4answers
79 views

non-trivial solutions to $x^2=x$ in a ring?

We know that when a ring is an integral domain, we have that: $$x^2=x \implies x^2-x = 0 \implies x(x-1) = 0$$ Since this is an integral domain, a product giving $0$ forces one of the the terms in ...
4
votes
1answer
160 views

To what is $X_{-m}+\alpha (a)Y_{-m}$ equal?

Let $F$ be an integral domain with characteristic $2$. Let $a\in F[t]$ and $a \notin F$. Let $\alpha (a)$ be a root of the equation $x^2+ax+1=0$. We define two sequences $X_m(a), Y_m(a) \in F[t], m ...
-1
votes
2answers
51 views

Prove isomorphism between two quotient modules [closed]

$R$ is an integral domain. For $x,y\in R$, with $x,y \neq0$, prove that $\dfrac{(x)}{(xy)} \cong \dfrac{R}{(y)}$ as $R$-modules. Do we need to find a mapping and the kernel and then use the first ...
1
vote
0answers
21 views

Characterization of valuation domains by means of their maximal ideal

I know the following theorem. Let $V, W$ be two valuation domains with the same fraction field $K$. Suppose $V$ and $W$ share the same maximal ideal. Then $V=W$. In other words, fixed a field ...
1
vote
1answer
47 views

Examples of Cohen-Macaulay integral domains

Question 1 Could you find a non Cohen-Macaulay ring $A$ without zero divisors. I would like $A$ to be as simple as possible. For instance, I want $A$ to be finitely generated alegbra over ...
5
votes
2answers
69 views

On an integral domain, with $(m,n)=1$, $a^m = b^m$ and $a^n = b^n$ implies $a=b$.

An exercise says: Let $R$ be an integral domain, let $m,n\in\mathbb{Z}$ such that $(m,n)=1$, then prove that $a^m = b^m$ and $a^n = b^n$ implies that $a=b$. I managed to prove it, but without using ...
6
votes
2answers
79 views

Show that an integral domain with finitely many ideals is a field

I know that an integral domain with finite number of elements is a field, but, how do relate this with the finitude of the number of ideals?
2
votes
1answer
26 views

Primary ideals in Prüfer domains

This is an exercise I found in a class test and I was struck trying to solve it. Let $D$ be a Prüfer domain (*) and let be $\mathfrak{q}_1,\mathfrak{q}_2$ two primary ideals of $D$. Then prove ...
4
votes
1answer
78 views

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
votes
1answer
143 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 $r\in R$. ...
1
vote
3answers
26 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 ...
0
votes
3answers
153 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. ...
0
votes
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 ...
-2
votes
2answers
121 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
votes
1answer
19 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.