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Questions tagged [ring-theory]

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

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1answer
15 views

Proving that if coprime $\alpha_{i}\in R$ divide b, then $\alpha_{1}…\alpha_{n}$ divide b.

Let $R$ be a principle ideal domain and let $\alpha_{1},...,\alpha_{n}\in R$ be such that $(\gcd(\alpha_{i},\alpha_{j}))=(1)$. Let $b\in R$ such that each $\alpha_{i}|b$. I want to show that $\...
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0answers
21 views

Endomorphism ring of $\mathbb{Z}/n\mathbb{Z}$. (in reference to elliptic curves over a ring)

I know for the group $E(\overline{\mathbb{F}_p})$ where $p$ is prime we have the Frobenius endomorphism, say $\phi$, satisfies the equation $(\phi^2 - [t]\phi + [q])P = [0]P$ in the endomorphism ring ...
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0answers
35 views

Reducibility Unnecessary Hypothesis?

Problem: Let $F$ be a field with $\text{char} F = p$ for some prime $p$. Show that if $X^p - X - a$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$ Solution in back of the ...
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1answer
47 views

Canonical ring structure on the tensor product $R \otimes_\mathbf{Z} S$.

Let $R$ and $S$ be commutative rings. I need to show that there is a unique ring structure on the $\mathbf{Z}$-module $T := R \otimes_\mathbf{Z} S$ such that $$ (r_1 \otimes s_1)(r_2 \otimes s_2) = ...
3
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2answers
17 views

Let $R$ be a ring with $1$. If there exist disjoint comaximal ideals $I, J$, then for any $R$-module $M$, $M = IM \oplus JM$.

This questions is from a past year paper. Let $R$ be a ring with $1$. Suppose there exist distinct ideals $I, J$ of $R$ such that $R = I + J$ and $I \cap J = \{ 0 \}$. (a) Show that there exists $a \...
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0answers
37 views

Modules, endomorphism and ideals

I was solving tasks in a book and there are two tasks where I have no idea how to solve them: 1) Let $R$ be commutative ring, $f: M\rightarrow N$ - isomorphism of $R$-modules. Prove isomorphism of ...
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1answer
76 views

Multiplication Table for $\Bbb Z_2[x] /<x^3 +x^2+x+1>$

The factor ring can be rewritten as $ \{a_o +a_1x+a_2x^2 \mid a_0,a_1, a_2 \in \mathbb{Z}_2 \}$. We figured out that the values on the multiplication table will be $\{0,1,1+x,1+x^2,x+x^2,x^2,1+x+x^2 \}...
1
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1answer
26 views

Proper Ideals with Norm Relatively Prime to Conductor

Let $K$ be an imaginary quadratic number field, and $\mathcal{O}_K$ the ring of integers. Let $\mathcal{O}$ be an order. Call the $\textit{conductor}$ $f = [\mathcal{O}_K:\mathcal{O}]$. Given some $\...
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4answers
62 views

Number Theory theorem regarding ring of integers in $\mathbb{Q}[\sqrt{D}]$

Here is the theorem that I need to prove For $K = \mathbb{Q}[\sqrt{D}]$ we have $$\begin{align}O_K = \begin{cases} \mathbb{Z}[\sqrt{D}] & D \equiv 2, 3 \mod 4\\ \mathbb{Z}\...
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1answer
38 views

Torsion Free Module over Dedeking Ring

Let $\phi: R \to A$ be a finite morphism of Dedekind rings (so $A$ is a finitely generated $R$-module) and $M$ a finitely generated $A$-module. Obviously, if we restrict the action of $A$ on $M$ to $R$...
4
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2answers
193 views

Determine the generator of an ideal of ring of integers

I am trying to find the generators of the ideal $(3)$ in the ring of integers of $\mathbb{Q}[\sqrt{-83}]$ the ring of integers is $\mathbb{Z}\left[\frac{1+\sqrt{-83}}{2}\right]$ I evaluated the ...
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1answer
22 views

Can anyone gives me an example of $V$ such that there exists at least one pair of elements $ a,c(\neq 0 ,1) \in D$ such that $V(a) > V(ac)$

$D$ is an Integral Domain. $V$ only satisfies the first Euclidean property, i.e. for all $\,a,b\in D\,$ if $\,b\neq 0\,$ then there are $\,q,r\in D\,$ such that $\, a = qb +r\,$ with either $r=0$ or $...
3
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1answer
33 views

If $R$ is a right-Noetherian ring and the Jacobson radical satisfies the right Artin-Rees property, then $\bigcap^{\infty}_{n=1}\text{Jac}(R)^n = 0$

A (two-sided) ideal $I$ of a ring with identity $R$ has the right Artin-Rees property if for any right-ideal $E$ of $R$, there exists an integer $n\geq1$ such that $E\cap I^n\subseteq EI$. If $R$ ...
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1answer
34 views

Can anyone give me an example of an Euclidean Domain $D$ whose Euclidean Valuation at a point $a$: $v(a)$ is bigger than $\min \{v(ax) : x\in D\}$?

Can anyone give me an example of an Euclidean Domain $D$ whose Euclidean Valuation at a point $a$ , $v(a)$ is bigger than $\min \{v(ax) : x\in D\}$ ? I know that second condition for being Euclidean ...
0
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1answer
19 views

Simple question about relative primes in entire rings.

Let $A$ be an entire ring. Let $a,b\in A$. Does The g.c.d. of $a,b$ is a multiplicative unit. $\Rightarrow$ $\langle a,b\rangle=A$ hold? If yes, how can I proof it?
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0answers
21 views

Smallest Ideal in $M_2(Z)$

What is the smallest ideal in $M_2(Z)$ containing $\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}$? I'm a bit unsure about what "smallest" means here. I've found all of the ideals of $M_2(Z)$, ...
1
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1answer
43 views

Is $\{p(x) ∈ \Bbb Q[x]\mid p(0) = 3\}$ an ideal of $\Bbb Q[x]$?

Is $\{p(x) ∈ \Bbb Q[x] \mid p(0) = 3\}$ an ideal of $\Bbb Q[x]$? I don't have any idea of how to start this problem. Any help would be great, thank you in advance!
0
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1answer
26 views

Divisor Properties

This is a general question regarding divisibility If $c|a$ and $c|b$, does $c|a+b$? Where $a,b,c$ are integers If yes, does this also hold for general rings?
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2answers
89 views

Adapting the Chinese Remainder Theorem (CRT) for integers to polynomials

I did a few examples using the CRT to solve congruences where everything was in terms of integers. I'm trying to use the same technique for polynomials over $\mathbb{Q}$, but I'm getting stuck. Here'...
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1answer
34 views

How to show that $x$ is irreducible in $\frac{\mathbb R[x,y]}{(x^2+y^2-1)}$?

How to show that $x$ is irreducible in $\frac{\mathbb R[x,y]}{(x^2+y^2-1)}$? I can show that x is not prime But unable to show x is irreducible. Please can any one help me to solve this problem ANy ...
1
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0answers
27 views

Quotient of Module over PID

Let $R$ be a PID Let $a$ be a nonzero element in $R$ Let $M=R/(a)$ For any p of R prove that $p^{k-1}M/p^kM\cong R/(p)$ if $k\leq n$ $p^{k-1}M/p^kM\cong 0$ if $k> n$ where $n$ is the power ...
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0answers
65 views

Find zero divisors for polynomials in several variables

I don't know how to find all zero divisors for polynomials in several variables. For example: $\mathbb{Z}_2[X,Y]/(X^2,XY,Y^2)\quad $ or $\quad \mathbb{Z}_4[X,Y]/(X^2,Y^2-XY)$ Can we to proceed ...
4
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2answers
41 views

About comment by Jacobson on proving that a morphism in $\mathbf{Ring}$ is monic iff it is injective

I am reading Nathan Jacobson's Basic Algebra II, Chapter 1 Categories, and in $\S$2 Some Basic Categorical Concepts, he introduces the notion of a morphism being monic or epic. He asks the reader to ...
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1answer
26 views

In a Euclidean ring $R$, prove $(a) ⊆ (b) \iff b|a$

Let $a, b$ be elements of a Euclidean ring $R$. Prove that $$(a) \subseteq (b) \iff b \;\text{divides}\;a.$$ I have no clue how to even start this. Any help would be great, thank you in advance!
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1answer
28 views

Proof of principle ideal

Let $I = \{p(x) ∈ \Bbb Z[x]: 5\mid p(0)\}$. Prove that $I$ is an ideal of $\Bbb Z[x]$ by finding a ring morphism from $\Bbb Z[x]$ to $\Bbb Z_5$ with kernel $I$. Prove that $I$ is not a principal ideal....
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1answer
40 views

Prove that $Z_{14}/(7)$ ≅ $Z_7$

Prove that $Z_{14}/(7)$ ≅ $Z_7$. I understand that I have to show that there exists a surjective function relating the two and use the morphism theorem, but I'm not sure how in this case. Any help ...
1
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1answer
51 views

Galois Theory for Finite Extensions of Rings

I am learning Galois theory for schemes from Lenstra's notes, and I have a question about how this might be phrased for integral extensions with a single generator. For fields, we have several ...
0
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1answer
18 views

Find all the ideals in the ring $Z_2$ x $Z_2$

Find all the ideals of $Z_2$ x $Z_2$ and all the ideals of $Z_7$. I understand how to find the ideals of a ring but I'm unsure of how to do it for rings of this small size. Any help would be great, ...
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0answers
14 views

If $f:R \rightarrow S$ is an onto ring homomorphism, $f(Z(R)) \subset Z(S)$ but the reverse inclusion is not always true.

If $f:R \rightarrow S$ is an onto ring homomorphism, I have just shown in an exercise that $f(Z(R)) \subset Z(S)$. I want to show this is not necessarily equality, i.e. two rings $R,S$ where there is ...
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3answers
38 views

In a ring $R$ with a $1_R$ with $u,v, u+v$ are units, show that $u^{-1} + v^{-1}$ is also a unit.

In a non-commutative ring $R$ with a $1_R$ with $u,v, u+v$ are units, show that $u^{-1} + v^{-1}$ is also a unit. The question seems relatively simple, but I'm having a tricky time explicitly ...
1
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1answer
29 views

Radicals and Generalized Eigenspaces

I am currently reviewing Jordan Normal Form. Say we have $T,$ a linear operator, on a finite-dimensional vector space $V.$ So if we consider an eigenvector $v$ with eigenvalue $\lambda,$ then our ...
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2answers
31 views

Prove the following ideal $I$ is not a principal ideal.

Prove the ideal $I = \left<X^2,3\right> \space $of$ \space \mathbb{Z}[X]$ is not a principal ideal. The solution I have been given is the following: Assume for contradiction that I were a ...
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0answers
21 views

Divisibility in Integers and General Rings [duplicate]

If x|y and y|x in integers, does this imply x=y? Similarly, for any general ring, if x|y and y|x in the ring, does this imply x=y?
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0answers
24 views

Double centralizer theorem for semisimple rings

Why every semisimple ring has the double centralizer property? Namely, let $R$ be a semisimple ring. We consider $R$ as left $R$-module. Let $E=End(R)$. Why every element in $End(R_E)$ is induced by ...
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0answers
22 views

Intersection with direct sum of modules [closed]

Let $R$ be an arbitrary ring and $M=\bigoplus_{i\in I} M_{i}$ be a direct sum of $R$-modules. Give an example shows that for a submodule $N$ of $M$ it is not necessary that $N=\bigoplus_{i\in I}N\...
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2answers
187 views

What is the Characteristic of a local ring? [on hold]

What is the Characteristic of a local ring ? We define Characteristic of a Commutative ring with $1$ say, $A$ in the following way: Define a ring homomorphism $\phi: \mathbb{Z} \to A$ by $\phi(n)=n \...
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0answers
45 views

Lipschitz primes

A Lipschitz integer is a Quaternion with integer coefficients. The norm is defined as $N(a+ib+jc+kd)=a^2+b^2+c^2+d^2$ which is a multiplicative function $N:\mathbb H\to\mathbb R$, $N(\alpha\beta)=N(\...
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1answer
13 views

To find annihilator of given module

I want to find annihilator of $Z_{14}$ and $Z_4 × Z_6$ as Z module. So in first case element a from Z will be in annihilator of $Z_{14}$ if 14 divides a,2a,3a,....,13a. But from here I am not getting ...
2
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1answer
11 views

Projection map for polynomial rings

Let $K$ be a field and Consider the projection map $\pi_{i,j} : K[X]/(X^i) \to K[X]/(X^j)$, for $j \leq i$. This is well-defined since $(X^i) \subseteq (X^j)$. I'm wondering what it looks like, is it ...
0
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1answer
18 views

Tower of ring extensions and flatness

Consider the following ring extensions: $$A\hookrightarrow\ B\hookrightarrow C$$ where we know that: $A$ is a complete DVR, $B$ and $C$ are Noetherian, Two-dimensional local rings and $B\...
1
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1answer
37 views

Given distinct nonzero elements $a$ and $b$ in $A$…prove $A$ has characteristic $x$.

I'm working on a problem from Pinter's Abstract Algebra and am wondering if someone can tell me if I'm on the right track. Let $A$ be a finite integral domain. Prove that if there are distinct ...
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1answer
40 views

How does the ring of algebraic integers in $\mathbb{Q}[\sqrt{D}]$ depend on $D\mod 4$?

Let $O_K$ be the ring of algebraic integers inside of $\mathbb{Q}[\sqrt{D}]$. Why is it that when $D \equiv 2, 3 \mod 4$ that $O_K = \{x + y\sqrt{D} : x, y \in \mathbb{Z}\}$ and that when $D \equiv 1 \...
1
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1answer
42 views

An ascending chain of prime ideals.

I am trying to get used to $\operatorname{Spec}$ of a ring. I know an example, when one prime ideal is contained in another for $\mathbb{C}[x,y]$. $(f) \subset (x-a,y-b)$, where $f(a,b) = 0$. Is ...
0
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1answer
28 views

When do tensor products of elements coincide

Let $M,N$ be $R$-modules and $m \otimes n, m' \otimes n' \in M \otimes_R N$ non-zero (EDIT) elements. When does $m \otimes n = m' \otimes n'$ hold? Obviously, this is true if either $(m',n)=(rm,rn')$ ...
6
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2answers
254 views

Computing $\phi(\frac32)$ where $\phi$ is an automorphism of $\mathbb Q[\sqrt2]$ such that $\phi(1)=1$ and $\phi(\sqrt2)=\sqrt2$

This question is a followup to this question about Field Automorphisms of $\mathbb{Q}[\sqrt{2}]$. Since $\mathbb{Q}[\sqrt{2}]$ is a vector space over $\mathbb{Q}$ with basis $\{1, \sqrt{2}\}$, I ...
0
votes
1answer
24 views

On natural homomorphism $\nu:R\longrightarrow S^{-1}R$

Let $R$ be a commutative ring with $1_R$ and $S$ an multiplicatively closed set. We define the natural homomorphism \begin{align*} \nu:R &\longrightarrow S^{-1}R, \\ a&\longmapsto \nu(a):=\...
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votes
0answers
37 views

How could I can prove this ring theory question? [duplicate]

If in a ring $R$ with unity, $(xy)^2 = x^2 y^2$ for all $x,y \in R$, then $R$ is commutative.
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0answers
62 views

Group of units of a field

So I'm a bit confused with this... I'm learning some field theory and I've just learnt about groups of units. With rings this makes sense. However say $F$ is a field then isn't every element in $F$ ...
2
votes
1answer
62 views

Functor of points of the completion of a ring

Let $R$ be a ring, with ideal $I$, and let $\widehat{R}_I$ be the completion $\varprojlim R / I^n$ of $R$. Can I somehow describe the functor $\mathrm{Hom}(\widehat{R}_I,?) : Rings \to Sets$ in an ...
2
votes
0answers
26 views

Prime property in noncommutative rings without identity

Let $R$ be a ring (without assuming identity or commutativity), and $P$ a proper ideal of $R$. Show that the following are equivalent: (a) For ideals $A,B$: $AB\subseteq P$ implies $A\subseteq P$ ...