This tag is for questions about rings, which are an algebraic structure studied in abstract algebra and algebraic number theory.

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2answers
104 views

$\mathbb{Z}_6[x]$/$\langle2x-3\rangle$ isomorphic to what?

To what common ring is $\mathbb{Z}_6[x]$/$\langle2x-3\rangle$ is isomorphic? Obviously not $\mathbb{Z}_6[\frac{3}{2}]$... Thanks!
0
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1answer
28 views

Ascending chain of ideals [duplicate]

Let $R$ be a commutative ring with identity such that every ascending chain of ideals terminate. Let $f:R \to R$ be a surjective homomorphism. Prove that it is an isomorphism.
7
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0answers
109 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
2
votes
2answers
47 views

Reducibility of a Cyclotomic Polynomial under the ring homomorphism $\mathbb{Z} \rightarrow \mathbb{F}_p$

I'm working through the following question: Question Reference: Oxford Part I Paper B2 2003 Find the monic polynomial $f \in \mathbb{Z}[X]$ whose roots are the complex primitive ...
0
votes
2answers
60 views

About the ring $\{a+b\sqrt{2}\mid a,b\in \Bbb Z\}$

Decide whether the $\{a+b\sqrt{2} \mid a,b \in \Bbb Z\}$ is a ring with usual addition and multiplication. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether ...
3
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0answers
41 views

Prime Ideals as Ring theoretic Ultrafilters

I am confused by the following statement in Awodey's Category Theory p. 35: Ring homomorphisms $A\to \mathbb Z$ into the initial ring $\mathbb Z$ play an analogous and equally important role [to ...
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0answers
40 views

How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

This question is quite closely related to my last question: Is it always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$? Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let ...
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0answers
11 views

A Ring with all Cyclic R-modules has Finite Projective Dimension but Infinite Global Dimension.

Can anybody give me an example of a ring $R$ with the property that each cyclic $R$-module has finite projective dimension even though the Global Dimension of $R$ is equal in infinity?
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1answer
21 views

Showing that $R$ is an ID with $0$ and $P$ only prime ideals.

I have a question which is as follows: Let $R$ be a unitary, commutative, local, noetherian ring with J principal. Prove that if $J$ is not nilpotent then R is an integral domain and that $0$ and ...
1
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2answers
37 views

Annihilator of $a'$ and $b'$ in the ring $\mathbb{Z}/(a'b')$ ?

I want to find the annihilator of $a'$ and $b'$ of the quotient ring $R=\mathbb{Z}/(a'b')$ where $a',\,b'>1$. So if I go by the definition, $ann(a')=\{r\in\mathbb{R}\mid ...
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0answers
12 views

What is the applications of number rings to structural/mechanical engineering?

What is the applications of number rings to structural/mechanical engineering? Or other related hardware stuff ordered by rings?
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0answers
16 views

… show that Noetherian domain

m is nonsquare integer with $m\equiv 1\pmod 4$ , prove that $Z+Z\left(\frac{1+\sqrt{m}}{2}\right)$ is Noetherian Domain. i was trying to show this domain is finitely generated but i ...
1
vote
1answer
25 views

Ring isomorphisms maintain noetherian property?

If a noetherian ring R is isomorphic to a ring S is S noetherian too? I am pretty sure that it isn't but can't find a counterexample Thanks!
0
votes
2answers
45 views

Problem related to prime ideals of B and A where B is integral over A

Let $ A $ be an entire ring, integrally closed. Let $ B $ be entire , integral over $A$. Let $ Q_1, Q_2$ be prime ideals of $B$ with $Q_1 \supseteq Q_2$ but $Q_1 \neq Q_2$.Let $P_i=Q_i \bigcap A$. ...
4
votes
4answers
97 views

Want to prove that some $\mathbb R[x]$-Module has no basis

So here is my question, Consider the $\mathbb R[X]$-module $\mathbb R[X,X^{-1}]$ i.e the $\mathbb R[x]$-module of all Laurent-Polynomials. I want to show that is module is not free i.e it has no ...
0
votes
1answer
33 views

Transfer Between LCM, GCD for Rings?

I am starting a chapter on divisibility in commutative rings, and I was wondering if there was a way to translate theorems about gcd to lcm and vice versa. I know the concepts are considered "dual" in ...
2
votes
1answer
41 views

Height of a specific maximal ideal

Let $k$ be a field, $k[x,y^2,xy,y^3]$ our ring and $\mathfrak a$ the ideal generated by $x,y^2, xy,y^3$. I want to determine the height $h(\mathfrak a)$ of $\mathfrak a$. My ideas: We see easily ...
0
votes
3answers
51 views

Show that the polynomial ring in $n-1$ variables is isomorphic to the polynomial ring in one variable $x_{n}$

For a ring $R$ and for $n \geq 1$, define $ S := R[x_{1},...,x_{n-1}]$ for the polynomial ring in $n-1$ variables with coefficients in $R$. Show that $R[x_{1},...,x_{n}]$ is isomorphic to the ...
2
votes
3answers
60 views

$\mathbb{Z}[i]/(1+i) \cong \mathbb{Z}/\mathbb{2Z}$

When I first looked at this problem, I thought that $\mathbb{Z}[i]/(1+i) \cong \mathbb{Z}/5\mathbb{Z}$, but apparently the correct answer is $\mathbb{Z}[i]/(1+i) \cong \mathbb{Z}/2\mathbb{Z}$. Here's ...
2
votes
1answer
50 views

Show that a set is a ring

Let $R\not=\{0\}$ be a commutative ring with unity. Let $I$ be a prime ideal in $R$. Let $S=R-I=\{x\in R|x\not\in I\}$. Let $F$ be a field that contains $R$ as a subring with the same unity. Show that ...
1
vote
1answer
52 views

Quotient Ring of finite order with root of irreducible polynomial

The question is as follows: Suppose that $\alpha\in\mathbb{C}$ is a root of the irreducible polynomial $f(t) = t^d + \sum_{i=0}^{d-1}a_it^i$ , where $a_i\in\mathbb{Z}$ ($0\le i\le d-1$). Let ...
0
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0answers
16 views

Exact Sequences of R-Modules

Here's a lemma in A Course in Ring Theory by Passman. In the proof it is mentioned, "But the kernel of the combined epimorphism $P\rightarrow B\rightarrow C$ is clearly equal to $E$". I don't ...
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1answer
36 views

Is every left maximal ideal the annihilator of a simple left module?

In my version of Noncommutative Algebra, by Benson Farb & R. Keith Dennis, in chapter I, section 2 on the Jacobson radical, it is claimed that … each maximal left ideal $I$ is the annihilator ...
0
votes
1answer
42 views

Definition of a field homomorphism

Given a field $F$ of characteristic zero, say $F=\mathbb{R}$, what is the minimal requirement for a function $\mu:F\to F$ to be a field homomorphism? (Do we need to require two axioms, one for ...
0
votes
1answer
34 views

Annihilator and Projective Dimension

I was reading the book A Course in Ring Theory by Passman and in it is the following lemma; and after this lemma there's a example which I don't quite understand; The main thing that I don't ...
0
votes
2answers
39 views

Is this set a subring of $\mathbb{Z}\times\mathbb{Z}$?

Is the set $S = \{(x,-x) : x \text{ is an integer}\}$ a subring of $\mathbb{Z}\times\mathbb{Z}$? I am not sure where to start here. Is $\mathbb{Z}\times\mathbb{Z}$ a matrix? It doesn't seem ...
1
vote
1answer
15 views

Projective Dimension and Supremum

Here is a lemma that appears in A Course in Ring Theory by Passman. In the last section of the proof the writer shows that, $\mbox{pd }A_i\leq n\iff \mbox{pd }A\leq n$ and finishes the proof. I don't ...
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votes
0answers
20 views

Definition of Subrings

(5)The set $\{[0], [2], [4]\}$ is a subring of $\mathbb Z(6)$. I bieleve this is false. It is closed under add/mult but does it have a 0 or an x where a + x = 0 is satisfied in S? What does it mean ...
3
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1answer
68 views

Rings (integral domain and fields)

True or false: (1) Every integral domain is a field (2) every field is an integral domain (3) the ring $\mathbb Z$ is a field. (4) the ring $\mathbb Z/(17)$ is a field. (5)The set $\{[0], [2], ...
2
votes
1answer
82 views

Is always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$?

Let $A$ be a commutative ring. Let $f \in A$. Let $A_f= A\left [ \frac{1}{f}\right ]$. Let $\hat{A}$ the $f$-adic completion of $A$. Is it always (even when $A$ is not noetherian) true that $$\hat{A} ...
0
votes
2answers
32 views

Let $R$ be a Noetherian ring. Then all finitely generated $R$-modules are Noetherian

Here is an excerpt of my lecture notes: " Claim I: Let $M$ be $R$- module and $N$ be submodule of $M.$ Then $M$ is Noetherian iff $N, \ M/N$ are Noetherian. Def: The ring $R$ is Noetherian iff the ...
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2answers
55 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 ...
4
votes
2answers
142 views

Can there be more ideals than elements of a ring?

Can there be more ideals than elements of a ring? This is related to my other question Having elements as Ideals . At first glance, it seems obvious that there would be less ideals than elements of a ...
1
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2answers
65 views

Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
1
vote
1answer
53 views

Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
0
votes
1answer
27 views

Why is $J/J^{2}$ is a vector space over $R/J$

Let $R$ be a commutative local ring with $1$ and let $J$ be the Jacobson radical. I have it written in my notes that $J/J^{2}$ is a vector space over the field $R/J$ that is an $R/J$ module. Why is ...
0
votes
1answer
28 views

Are these polynomials irreducible over $\Bbb Z$?

I have received these problems and I'm not sure where to start: Are these polynomials irreducible over $\Bbb Z[X]$ ? a) $X^{65536} + 1$ (which is really (c) for n = 16) b) $X^{10} + X^9 + X^8 + ...
0
votes
1answer
26 views

Proving $(φ(x)\cdot ψ(x)) \cdot ω(x)=φ(x) \cdot (ψ(x)\cdot ω(x))$ where $φ,ψ,ω$ are polynomials on a ring $R[X]$

If I take $3$ random polynomials $φ,ψ,ω$ on a ring $R[x]$, I'm trying to prove associativity which is very obvious. But I have trouble on the algebra part with the sums. I know that given $2$ ...
0
votes
1answer
23 views

A problem on $\text{ACCP}$

Let $R$ be a commutative ring. Could anyone advise me on how to prove $R$ has $\text{ACCP}$ (Ascending chain condition for principal ideals) iff every collection of principal ideals of $R$ has maximal ...
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2answers
27 views

Proving C is a Subring of R

For the three axioms Is $0$ contained in C? I got that by putting $a=0$ $(0)(r)=(r)(0) = 0$ For is $a-b$ contained in $C$ and Is $(a)(b)$ contained in $C$ I' ve been playing around with the ...
0
votes
1answer
20 views

If $R$ is $\text{UFD},$ then $R[X,Y]$ is $\text{UFD}.$

Let $R$ be commutative ring with $1.$ Suppose $R$ is $\text{UFD}.$ Could anyone advise me on how to prove $R[X,Y]$ is $\text{UFD}\ ?$ Thank you.
0
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1answer
63 views

Topics on Algebra-Ring Theory for an essay-project on undergraduate level!!

I'm an undergraduate in the mathematics field. So i want to be a little more productive and wanted to do an essay or project on Algebra Ring-Theory (because I got it as a course this semester -Ring ...
1
vote
1answer
31 views

Properties of GCD in rings

Let $R$ be subring of integral domain $S.$ Suppose $R$ is $\text{PID}.$ Let $a\in R$ be a greatest common divisor of $r_1,r_2$ in $R$. ($r_1,r_2 \in R$, not both zero). Could anyone advise me on how ...
1
vote
2answers
61 views

algebraic integer $\alpha$ + polynomial relation $\beta$ and $\alpha$ $\Rightarrow$ $\beta$ algebraic integer.

Assume $\beta$ can be expressed in terms of polynomial relation in $\mathbb{Z}[\alpha]$. Where $\alpha$ is an algebraic integer (i.e. $\alpha$ is the root of a polynomial in $\mathbb{Z}[X]$. How can ...
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0answers
26 views

some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
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2answers
61 views

Question concerning $M_1\cap K=M_2\cap K$ and $M_1+K=M_2+K$

I have a ring $R$ with $K\le M$ and submodules $M_1,M_2$. If we have that: $$M_1\cap K=M_2\cap K \text{ and } M_1+K=M_2+K$$ can we conclude that $M_1=M_2$? I don't think that this is true ...
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votes
2answers
73 views

Is there a ring so that any two distinct non-zero elements do not commute?

I was wondering if there was a ring so that any two distinct non-zero elements do not commute. Formally, is there a ring $R\not=\{0\}$ so that $$\forall x,y\in R\setminus\{0\}, x\not= y\implies ...
4
votes
0answers
49 views

The ring of homogeneous polynomials

I think I found an error in my textbook, but I am not completely sure. The book is Hulek, Elementary algebraic geometry, pag. 73. There is a theorem showing that $U_i$ and the affine space ...
4
votes
1answer
50 views

Maximal ideals in the ring of eventually constant sequences of real numbers

For homework I am studying the ring $R$ of eventually constant sequences of real numbers (with multiplication and addition defined componentwise). What are the maximal ideals of $R$? By looking at ...
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vote
2answers
33 views

Associates in Domains

Let D be a domain and $a, b \in D^*$. Show that $a$ is a proper divisor of $b$ if and only if $b=ax$ for some nonzero nonunit $x$. I'm just really not sure how to start this. Any advice would be ...