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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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If $Q$ is a primary ideal of the ring $R$, then its nil-radical $\sqrt{Q}$ is a prime ideal.

The book I'm reading assumes commutativity as it began with $ab \in \sqrt{Q} \Rightarrow (ab)^n=a^nb^n\in Q$. I don't understand why. I attempted writing my own proof. Please tell me my mistakes. ...
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An ideal $I$ of the ring $R$ is a semiprime ideal if and only if the quotient ring $R/I$ has no nonzero nilpotent elements.

Proposition: An ideal $I$ of the ring $R$ is a semiprime ideal if and only if the quotient ring $R/I$ has no nonzero nilpotent elements. Proof?: Let $I$ be an ideal of $R$ s.t. $I=\sqrt{I}$; that is, ...
5
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1answer
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When can we “reduce” $(n\Bbb Z/m\Bbb Z)$?

$(3\Bbb Z/6\Bbb Z) \cong (\Bbb Z/2\Bbb Z)$ can be easily shown by using the first isomorphism theorem, but I heard that we cannot say that $(2\Bbb Z/6\Bbb Z) \cong (\Bbb Z/3\Bbb Z)$. Why not? And ...
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0answers
34 views

Existence of $(a,b)\in\mathbb{C}^2$ satisfying $(f)\cap\mathbb{Q}[x,y]=(x-a,y-b)\cap\mathbb{Q}[x,y]$ for irreducible f $\in \mathbb{C}[x.y]$

Prove that, if $f$ is an irreducible element in $\mathbb{C}[x,y]$, then there exist $(a,b)\in\mathbb{C}^2$ such that $(f)\cap\mathbb{Q}[x,y]=(x-a,y-b)\cap\mathbb{Q}[x,y]$. The original question is, ...
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Specifying a base $B_R$ for the row space and a base $B_C$ for the column space.

The following matrix over $\mathbb{Z_5}$ is given: $$ \begin{bmatrix} 2 & 2& 2& 3\\ 1&3&1&3\\ 3&0&2&2\\ 4&1&0&4\\ 1&2&2&0 \end{bmatrix} $$...
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3answers
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Why is $F[x]/(x^n)$ a local ring?

How is $\frac{F[x]}{(x^n)}$ a local ring? I was trying to show the elements which are not units are nilpotent. But not being able to prove it properly. Please give some hint.
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2answers
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GCD of $f=X^3 +9X^2 +10X +3$ and $g= X^2 -X -2$ in $\mathbb{Z}/5\mathbb{Z}$.

I have to calculate the gcd of $f=X^3 +9X^2 +10X +3$ and $g= X^2 -X -2$ in $\mathbb{Q}[X]$ and $\mathbb{Z}/5\mathbb{Z}$. In $\mathbb{Q}[X]$ I got that $X+1$ is a gcd and therefore $r(X+1)$ since $\...
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1answer
49 views

Show that $\frac{x(x-1) \dots (x-n+1)}{n!} \in \mathbb{Z}$ with $x \in \mathbb{Z}$ [duplicate]

Problem: Let polynomial $Q_n (x) = \frac{x(x-1) \dots (x-n+1)}{n!} \in R[x]$ for some ring $R$. Show that $\forall t \in \mathbb{Z}, Q_n (t) \in \mathbb{Z}$. My solution: For each $t \in \mathbb{Z}$, ...
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0answers
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What was the motivation behind the fraction ideal's name?

The fraction ideal is the ideal $$(I:J)=\{x\in R:xJ\subseteq I\}$$where $I, J$ are ideals in $R$. My question is, why is it called a fraction ideal? (Sometimes it is called the "Ideal quotient of $I$ ...
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2answers
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Let $ R$ be a commutative ring with $1$ and $S$ be a subring of $R$ containing $1$.

Let $ R$ be a commutative ring with $1$ and $S$ be a subring of $R$ containing $1$. Can we say $R=S$ always ?
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1answer
40 views

Localization at annihilators of an ideal

I was reading this post and on line +10-11 of the proof of lemma 27.25.1, it seems to claim the following: Let $A$ be a ring, $I \subseteq A$ an ideal, and $M$ an $A$-module. Let $M_I:=\{ x \in M\...
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1answer
46 views

coprime ideals in a ring

Suppose $R$ is a ring ($R$ may not have a unit and can be non-commutative), $I,J$ are two nonzero proper ideals in $R$ such that $I+J=R$ and $I\cap J\neq 0$. I wonder if there exists a possibility ...
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27 views

Generators of localization of an order with prescribed norm

I've been thinking about various variants of this problem for about 2 weeks, and I can't help but think there's some simple thing I'm missing. Consider the imaginary, quadratic order $\mathcal{O} = \...
2
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1answer
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$\varphi : R\rightarrow \mathbb{Z}/p\mathbb{Z},x\mapsto x+p\mathbb{Z}$ is well-defined and surjective.

Given the ring $$R=\{x\in\mathbb{Q}\mid x=\frac{a}{b}, a.b\in\mathbb{Z}, p\nmid b\}$$ for a prime number $p$. Now I have to show that $$\varphi : R\rightarrow \mathbb{Z}/p\mathbb{Z},x\mapsto x+p\...
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2answers
22 views

Finding generator for the ideal generated by $a$ and $b$ in an euclidean domain

Let $D$ be a euclidean domain and $a, b \in D$. Show that $M = \{xa + yb \ \mid \ x, y \in D\} $ is an ideal of $D$. Find $d \in D$ such that $M = \langle d \rangle$ and prove your claim. My effort: ...
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1answer
52 views

Is $\mathbb{Z}[x]$ an integral domain? If so, why? [duplicate]

I'm trying to solve a larger problem about maximal and prime ideal, and knowing if $\mathbb{Z}[x]$ is an integral domain would really help me
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2answers
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For prime quadratic integer $\pi$, $x \equiv 1$ (mod $\pi$), Show $x^2 \equiv1$ (mod $\pi^2$) and $x^3 \equiv 1$ (mod $\pi^3$) is not always true.

I was working through one of the problems given to me on my problem set for a number theory class and I would like some help in an attempt to learn. Could someone help me with the following question? ...
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9 views

left and right regularity in relation to mono and epi?

Consider a category $C$. Suppose $f:a\to b$ is monomorphism. Then for any $g,h:c\to a$ s.t. $fg=fh\implies g=h$. Similarly, one dualizes the statement to obtain the statement for epimorphism. $\...
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3answers
30 views

Prove The Derivative Rules in the Ring of Polynomials

Let R be a commutative ring with unity element 1. Let $f(x)\in R[x]$ and define its derivative as $f'(x)=r_1 +2(r_2)x+...+n(r_n)x^{n-1}$. Prove that $(f+g)'(x)=f'(x)+g'(x)$ and that $(fg)'(x)=f'(x)g(x)...
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0answers
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Compatibility with multiplication of a cyclic order on a ring

Considering a linear order on the additive group of a ring is compatible with multiplication if: $a < b \implies ax < bx$ and $xa < xb$ for any positive $x$, we could define compatibility ...
0
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1answer
24 views

Showing that $\varphi:\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Z}/m\mathbb{Z}$ is a well-defined surjective ring homomorphism

I have to show that $$\varphi:\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Z}/m\mathbb{Z},a+n\mathbb{Z}\mapsto a+m\mathbb{Z}$$ is a well-defined and surjective ring homomorphism for $m|n$. My idea was ...
0
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2answers
35 views

Is $R^\infty$ a ring?

Let $(R,+,\cdot,0,1)$ be a ring, and consider the set $$R^\infty=\left\{\{a_n\}_{n=1}^\infty:a_k\in R\text{ for all } k \in \mathbb{Z}_{>0}\right\}$$ with operations $\oplus$ and $\odot$ on $R^\...
0
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1answer
43 views

Is a finitely generated module over the field of fractions is also finitely generated over the original integral domain?

Let $R$ be an integral domain and $F$ its field of fractions. Let $M$ be a finitely generated $F$-module. Question: Is $M$ also a finitely generated $R$-module? I know that $M$ is an $R$-module ...
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2answers
61 views

$\sqrt{180\mathbb Z} = 30\mathbb Z$, $(180\mathbb Z:700\mathbb Z)= 9\mathbb Z$

(First time studying rings, and I need some help on this example about radical ideal and fraction ideal) Let $\sqrt{I}$ be the radical ideal on the commutative ring $R$, defined as $\sqrt{I}=\{r\in R:...
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Why is the normalisation of a singular curve never flat?

In Vakil's notes on Algebraic Geometry, he states in exercise 24.4.H that the normalisation of a singular curve is never flat and he claims that this is a simple consequence of the fact that every ...
0
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1answer
26 views

reference for theorem commutative ring R is semi simple iff direct product of fields

Can I get any reference(book or journal paper) for the theorem, Let R be a commutative ring, R is semisimple if and only if it is isomorphic to a direct product of a finite number of fields.
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Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal?

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal? Justify your answer. I am a bit hesitant about asking this here. The question is not "How to Solve This Problem". ...
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0answers
39 views

Ring homomorphisms from $\mathbb{R}$ to another unital ring $S$.

We know that there is only one non-trivial ring homomorphism from $\mathbb{Z}$ or $\mathbb{Q}$ to another unital ring $S$. What’s more,when we consider the automorphism of $\mathbb{R}$,it is unique ...
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Annihilator and maximal ideal in finite ring

I have this proposition Let $R$ be a finite commutative ring with unity. If $M$ is a maximal ideal in $R$ then $\exists m\in M: M=Ann(m)$ I do not know how to give this $m$ and why the ...
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18 views

Prove that the following conditions are equivalent for a $_RP$ projective module

Let $_RP$ be a projective module, then: End($_RP$) is semiperfect P is semiperfect and finitely generated are quivalent. I have to prove this, but I think I'm not understanding the idea behind. ...
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0answers
47 views

$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$

$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$ I think $p$ is supposed to be a prime for the only ...
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3answers
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Showing that $\sqrt{5} \in \mathbb{Q}(\sqrt[p]{2} + \sqrt{5})$

I am attempting to show that $\sqrt{5} \in \mathbb{Q}(\sqrt[p]{2} + \sqrt{5})$, where $p > 2$ is prime. I have already shown that $[\mathbb{Q}(\sqrt[p]{2}, \sqrt{5}) : \mathbb{Q}] = 2p$. If needs ...
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2answers
25 views

Image of an ideal under a surjective ring homomorphism is an ideal

Let $\phi: R \longrightarrow R'$ be a surjective ring homomorphism and $I$ an ideal in $R$. Show that $\phi(I) = \{ \phi (r) : r \in I \}$ is an ideal in $R'$. So I asked this question a couple ...
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2answers
28 views

Describe the set of quadratic integers α in Q[sqrt−3] for which α ̄ and α are associates.

I was working through some textbook problems for my Number Theory class and needed some help with the following question: Describe the set of quadratic integers α in Q[sqrt−3] for which α ̄ and α are ...
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2answers
27 views

Isomorphism between C and another ring

Let the operations of addition and multiplication on the set $K = {at+bu : a,b ∈ R}$, where $t$ and $u$ are formal symbols, be defined as follows: $(at+bu)+(ct+du) = (a+c)t+(b+d)u$, $(at+bu)·(ct+du) ...
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1answer
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Two questions on the ring S := {a+bs : a,b ∈ R}

Let $s$ be a formal symbol. Define addition and multiplication operations on the set $S := {a+bs : a,b ∈ R}$ (with curly brackets) by the rules $(a+bs)+(c+ds) := (a+c)+(b+d)s$, $(a+bs)(c+ds) := (ac+...
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1answer
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Number of ring homomorphism from $\Bbb Z[x,y]$ to $\Bbb F_2[x]/(x^3+x^2+x+1)$

Find the number of ring homomorphism from $\Bbb Z[x,y]$ to $\Bbb F_2[x]/(x^3+x^2+x+1)$. Now I have observed that $(x^3+x^2+x+1)=(x+1)^3$ in $\Bbb F_2[x]$. Then $\Bbb F_2[x]/(x^3+x^2+x+1)=\Bbb F_2[x]/(...
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1answer
77 views

Irreducibility of $t ^ { 4 } + t - 1$ over $\mathbb{F}_{25}$

Problem: Let $f ( t ) = t ^ { 4 } + t - 1 \in \mathbb { F } _ { 5 } [ t ]$, show that $f(t)$ has no roots in $\mathbb{F}_{25}$. I tried the following: suppose $f(t)$ has a root in $\mathbb{F}_{25}$, ...
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0answers
37 views

Let $K$ be a ring. Then $K[X_1,X_2] \cong K[X_1][X_2]$

This is a lemma from textbook Analysis I by Amann/Escher. I present my attempt below. Does it look fine or contain gaps/errors? Thank you for your verification! Let $K$ be a ring. Then $K[X_1,X_2] \...
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0answers
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Construction of Roots of Polynomials [on hold]

I'm wondering if I can construct a root of the following three polynomials: $x^2-7x-13$ $x^8-16$ $x^4+x^3-12x^2+7x-1$ I think I can because the field extension of a root and Q is always divisible by ...
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0answers
101 views

show $\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain

I've got to show that $A:=\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain I know that $A$ is isomorphic to $\mathbb{C}[t,t^{-1}]$ and that this a subfield of $\mathbb{C}[t]$ which is a PID. So ...
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0answers
9 views

Computing a quotient ring. [duplicate]

I wonder how to compute for some $m,n > 0$ the ring $$ (\mathbb{Z}/m\mathbb{Z})/(\overline{n}). $$ I believe it should be the case that $$ (\mathbb{Z}/m\mathbb{Z})/(\overline{n}) \cong \mathbb{Z}/...
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0answers
20 views

The class of left serial rings is closed under extensions

A class $S$ is closed under extension if given an ideal $I \subseteq R$ such that $I\in S$ and $R/I\in S$, then $R\in S$. A ring $R$ is left serial if it is a direct sum of left uniserial rings. ...
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0answers
23 views

Suggestion for seminar about rings of continuous functions [closed]

I have to do a seminar about the rings of continuous functions, it will be a part of a course in topology. The main topic of my seminar will be the functor from the topological space and the ...
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0answers
42 views

Embedding a polynomial ring into $\mathbb{Z}_{n^s}$

My setting is the following: $n$ is a product of two big primes (RSA-like), and I am given a $R = \mathbb{Z}/n^s\mathbb{Z}$ as a space to work with. I would like to represent elements of $R$ as ...
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0answers
47 views

An example of an argument using generic points to prove a “closed” condition

Every irreducible affine scheme $\mathrm{Spec}(R)$ contains a generic point, namely $\eta:=\mathrm{Nil}(R)$. If $R$ is a domain then $\eta=(0)$. This is a point which is Zariski dense in $\mathrm{Spec}...
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1answer
26 views

Is every point in the spectrum of a ring $R$ closed?

I am just getting started on spectrums of rings. I see how it is natural to augment the set of prime ideals with the Zariski topology, but from my poor intuition on the topic I don't see how any of ...
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1answer
43 views

Is there a ring whose total ring of fractions is not a field?

I am trying to come up with an example of a ring whose total quotient ring is not a field. I know that if $R$ is a domain, then every total quotient ring has to be a field, however in the general case ...
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0answers
25 views

What's a finite non-commutative ring? [duplicate]

Please give me some examples, this is an exam question of a 1st year math module.
0
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3answers
35 views

Show that $x^2 +1$ is irreducible in $\mathbb{R}[x]$, but it has roots in $\mathbb{R}[x]\space/\space(x^2 +1) \cong \mathbb{C}$ [duplicate]

So I know that for something to be irreducible, then it cannot be written as the product of non-constant polynomials of smaller degree, but I don't know how to show that the factors don't exist is the ...