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
351 views

Example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor

I don't remember whether there was a special name for a commutative ring where every non-invertible element is a zero-divisor. And I also forgot the different ways in which a non-invertible element ...
2
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3answers
71 views

Are these two rings isomorphic?

I have these two rings $k[x,y,1/y]/(x^2 +1 - y^2)$ and $k[u,v,1/v]/ (u^2 + v^2 -1)$, where $k$ is a field, and I was wondering if these two rings were isomorphic or not. I would greatly ...
3
votes
1answer
76 views

Representatives of simple modules

For each of the following rings find a list of representatives of all simple $R$-modules: 1) $R=\mathbb C[x]$ 2) $R=\mathbb R[x]$ What I've tried was: I know that $M$ is a simple ...
2
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0answers
51 views

Dixmier Conjecture

In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Which are some consequences of Dixmier conjecture ...
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4answers
105 views

If every free $R$-module has the property that independence implies extendibility, is $R$ necessarily a field?

Definition. Whenever $M$ is a free $R$-module, let us call a subset $A$ of $M$ extendible iff there is a basis $B$ for $M$ such that $A \subseteq B$. (Is there a standard name for this condition?) ...
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0answers
89 views

Surjectivity of a Free Module Homomorphism implies Injectivity?

I am trying to prove the following: Let $R$ be a commutative ring and $M$ be a free $R$-module having a finite basis of $n$ elements. Let $T:M\to M$ be a surjective $R$-module homomorphism. ...
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1answer
31 views

Division in $\mathbb{Z}_7 [x]$

I am trying to perform division in $\mathbb{Z}_7 [x]$. I want to divide $f(x)=2x^4+x^2-x+1$ by $g(x)=2x-1$. I end up with $f(x)=g(x)\times (x^3 + \frac{1}{2}x^2 ...
0
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1answer
101 views

Property of prime ideals of $\Bbb{Z}[X_1,…,X_n]$

Let $P$ be a prime ideal of $\Bbb{Z}[X_1,...,X_n]$. How to show that there exist a prime number $p$ such that $(p)+P$ is not $\Bbb{Z}[X_1,...,X_n]$.
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0answers
110 views

Hartshorne, Exercise 3.18, Chapter 2

Let $B$ be a noetherian integral domain, let $A$ be a subring of $B$ such that $B$ is a finitely generated $A$ algebra. Assume that $A$ is also noetherian. Let $b$ be a non-zero element of $B$. How ...
1
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1answer
104 views

Which associative and commutative operations are defined for any commutative ring?

In a commutative (unital) ring $R$, the binary operation $x*_{abc}y:=axy+b(x+y)+c$ can be defined for arbitrary $a,b,c\in R$. This operation is obviously commutative. It is associative iff ...
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1answer
27 views

A question about Rings (invertible) [duplicate]

Let R be a ring. Suppose that there exist an element r ∈ R with r^n =0, for some n ≥ 1. Prove that 1-r is invertible.
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2answers
63 views

Fibers of $\operatorname{Spec}(R)\to\operatorname{Spec}(S):\mathfrak{q}\mapsto \mathfrak{q}\cap S$ are discrete?

Suppose $S$ is a subring of a commutative ring $R$, such that $R$ is finitely generated as an $S$-module. I"m curious about a property of the map ...
2
votes
2answers
164 views

Associates in the ring of continuous real-valued functions on $[0,1]$

I have tried to give a proof of the following theorem but I feel very unsure and would be very grateful if someone would check it for me Many thanks! Theorem. Let $R$ be the ring $C[0,1]$ of ...
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2answers
53 views

Is the length of a module over a simple artinian ring an invariant?

If $R$ is a simple artinian ring, Wedderburn theory tells us that $R=Mat_n(D)$ for some $n\geq 1$ and division ring $D$ and also every $R$-module $M$ is a direct sum of finitely many copies of the ...
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votes
3answers
39 views

Checking irreducibility of $3x+6$ in $\mathbb Q[x]$ and $\mathbb Z[x]$

Any hint How should I check whether $3x+6$ is irreducible in : 1.) $\mathbb Q[x]$ 2.) $\mathbb Z[x]$
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1answer
36 views

Associates in ring of polynomials

I can't understand what actually being Associate means in Rings of Polynomials. The book states: Two elements $a$ and $b$ of a commutative ring with unity are associates if there exists a unit ...
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2answers
352 views

Embedding a ring into a ring with unity

I was reading the theorem on Embedding of a ring into a ring with unity which is as follows: Let R be ring and $R\times \mathbb Z=\{(r,n)|r\in R,n\in \mathbb Z\}$ . this is a ring with ...
2
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1answer
45 views

Proving Irreduciblity in Polynomial Quotient Rings

I'm working on an exercise from Dummit and Foote, and I've gotten down to the following lemma that makes everything I need work out, the only problem is that I'm not sure how to prove it (or whether ...
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1answer
49 views

Reference on Modules

I'm looking for a book that provides a nice introduction to Modules for a student that already had a first course in Abstract Algebra in groups and rings. The book should explain why are modules ...
3
votes
1answer
76 views

How to write $210\in\mathbb Z[\sqrt{-1}]$ as a product of prime elements?

How can I write $210\in\mathbb Z[\sqrt{-1}]$ as a product of prime elements? Progress I factored $2\in\mathbb Z[i]=(1-i)(1+i)$ and $5\in\mathbb Z[i]=(2-i)(2+i)$. I cannot do it for $3$ and $7$ ...
1
vote
1answer
82 views

Smallest subring containing $\sqrt{5}$

I want to find the smallest subring of $\mathbb R$ which contains $\mathbb Q$ and $\sqrt 5$. I am sure that$\{a+b\sqrt{5}:a,b \in \mathbb Q \}$ is the right candidate. I already showed that this is ...
0
votes
1answer
64 views

A question on non commutative ring or algebra

Assume that $R$ is a ring such that $R=I+J$ where $I$ and $J$ are 2 -sided ideal.(This is not a direct sum) If $I$ and $J$ are commutative does it implies that $R$ is a commutative ring? Please ...
3
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2answers
91 views

Good reference book for quadratic integer rings?

Could anyone direct me to a good reference book(s) for quadratic integer rings? Ideally, the reference would begin with their elementary properties and then proceed through their ring-theoretic ...
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2answers
78 views

Maximal ideal in $\mathbb{Q}[x,y]$

I am trying to prove that $(x,y)$ is a maximal ideal of $\mathbb{Q}[x,y]$. Since an ideal $I \subseteq R$ is maximal if and only if $R/I$ is a field, it suffices to prove that $\mathbb{Q}[x,y]/(x,y)$ ...
0
votes
1answer
60 views

Simple question on tensoring by a quotient ring

$A \subset B$ is an extension of commutative rings s.t. $B$ is a f.g. free $A$-module of rank $n$, so I have $A^n \stackrel{\sim}{\longrightarrow} B$ as $A$-modules. Let $\mathfrak a$ be an ideal of ...
3
votes
1answer
71 views

Simple Maximal Ideal Question.

Question: Let $R = \{a + bi | a,b \in \Bbb Z \}$, let $M = \{x(2 + i) | x \in R\}$. Prove M is a maximal ideal of R. I just started learning about ideals so I apologize for asking a basic question, ...
4
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1answer
314 views

why $\mathbb Z[\sqrt 2] \ncong \mathbb Z[\sqrt 3]$?

The following is a question from section $3.11$ of the book An introduction to abstract algebra by Allenby: Explain intuitively why $\mathbb Z[\sqrt 2] \ncong \mathbb Z[\sqrt 3]$.back your ...
0
votes
1answer
40 views

In $\mathbb Z[x]$, show that the only common divisors of $2$ and $x$ are $1$ and $-1$.

In $\mathbb Z[x]$, show that the only common divisors of $2$ and $x$ are $1$ and $-1$. $\mathbb Z[x]$ is the ring of poloynomials with integer coefficients. This should be a pretty trivial question. ...
0
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1answer
49 views

Ideals and product of ideals

If $I$ is an ideal of a ring $R$, what is the meaning of $I^k$? 1) Is it the collection of $k-$ tuples of elements of $I$? 2) Or is it the collection of finite sums of $k$ products of elements of ...
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1answer
43 views

The fundamental unit in the ring of algebraic integers. 1

Let $R$ be a ring. Suppose that there exists an element $r\in R$ with $r^n = 0$, for some $n \geq 1$. Prove that $1 - r$ is invertible. May I know how we can prove this theory with some examples?
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2answers
59 views

In $\mathbb{Z}_q$ where $q$ is prime, show that $[a^q]=[a]$ for all $[a]\in \mathbb{Z}_q$

Question: In $\mathbb{Z}_q$ where $q$ is prime, show that $a^q=a$ for all $a\in \mathbb{Z}_q$. My attempt: To show $[a^q]=[a]$ for all $[a]\in \mathbb{Z}_q$, it suffices to show that for any ...
0
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2answers
38 views

Show that $\deg(fg) = m+n$

Let $R$, a ring with a $1$ and $f,g$ two polynomials, where $\deg(f)=n, \deg(g)=m$. Also, there's a $c\in R$ such that $b_mc = 1$. Show that $\deg(fg)=m+n$. I'd be glad for a guidance. Thanks
1
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1answer
39 views

Let $I$ be a proper ideal of a ring $R$. Then $IR[\alpha_1, … , \alpha_n]$ is a proper ideal of $R[\alpha_1, … , \alpha_n]$

Let $I$ be a proper ideal of the commutative ring $R$. Then $IR[\alpha_1, ... , \alpha_n]$ is a proper ideal of $R[\alpha_1, ... , \alpha_n]$ I thought of using the fact that an ideal of any ring ...
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3answers
68 views

Idempotents in $\mathbf{CRing}$

I'm not able to find an example of an idempotent morphism different from an identity in the category of commutative rings with unity (an idempotent, as a morphism in that category, must preserve 1, ...
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1answer
597 views

list the distinct principal ideals in $\mathbb{ℤ}_2 \times \mathbb{ℤ}_3$

How do I find and list the distinct principal ideals in ℤ2xℤ3? I know that Z2 has 0,1 and that Z3 has 0,1,2, but I'm not sure how to list them and how to find ideals in Z2xZ3
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0answers
68 views

(Updated): Finding the kernel of a ring morphism

Let $m,n \in \mathbb{Z} \setminus \lbrace 0 \rbrace $, consider $$\varphi: \begin{cases} \mathbb{Z}_{/<m \cdot n >} &\longrightarrow \mathbb{Z}_{/<m>} \times ...
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1answer
54 views

Prime ideals in non-commutative ring

On Wikipedia it says that an ideal $I \neq R$ in a non-commutative ring $R$ is prime if whenever two ideals $A,B$ satisfy $AB \subseteq I$ then either $A \subseteq I$ or $B \subseteq I$. It also ...
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3answers
146 views

In a noncommutative ring, is there always a pair $x,y$ such that $xy-yx=1$?

Let $R$ be a non-commutative ring. Are there two element $x,y\in R$ such that $xy-yx=1_{R}$? I have proved it is true for $R$ being an algebra with finite dimension. Sorry, I made a mistake, should ...
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votes
2answers
144 views

Does every infinite field contain the integers as a subring?

I simply ask because if $1+1=2(1)=2$ then this would imply that all positive integers are contained, and as every element in a field has a negative all the negative integers are contained. At the same ...
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1answer
49 views

Questions Regarding a Ring

I am extremely new to abstract math. I was given the following problem and below each of the questions, I have my answer. I can't imagine it is right because I am so confused. Please point me in the ...
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1answer
46 views

Ring/Nilpotent Proof [duplicate]

Let $R$ be a ring with unity, and suppose $x\in R$ is nilpotent $(i.e. x^n=0$ for some positive integer $n$ $)$. Prove that $1-x$ is a unit in $R$. Any hints or proofs are greatly appreciated. Rings ...
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2answers
59 views

Finding roots of $x^9 + 1$ modulo $19$

As part of a problem to factorise $f = x^6 + x^3 + 1$ over $\mathbb F_{19}$, I've realised that $f$ is a factor of $x^{18} - 1 = (x^9 + 1)(x-1)(x^6 + x^3 + 1)(x^2 + x + 1)$ which splits into linear ...
0
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1answer
114 views

Nilpotent and invertible polynomials over noncommutative rings

Let $R$ be a noncommutative ring. 1) Prove or disprove: $a_0+a_1 x+\cdots+a_n x^n\in R[x]$ is nilpotent iff $a_i$ is nilpotent $\forall i$. 2) Prove or disprove: $a_0+a_1 x+\cdots+a_n x^n\in R[x]$ ...
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1answer
63 views

An equivalent condition with $\{0\}$ being the only nilpotent ideal

In a ring $R$ prove that $\{0\}$ is the only nilpotent ideal if and only if for every ideals $A$ and $B$ from $R$, $AB=\{0\}$ implies $A\cap B=\{0\} $.
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2answers
142 views

Doubt regarding Sum of two subrings need not be subring.

We know that sum of two subrings need not be a subring ,but then why is the following so: Let $A$ be a subring of a ring $R$ and $I$ an ideal of $R$ . Then $A+I=\{a+i|a\in A,i\in I\}$ is a ...
3
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1answer
104 views

Self-learning Book recommendation for topics in ring-theory

I failed badly in my Internal examination in ring theory , and at any cost want to improve upon my grades in the final eamination,with a month and a half to go .... I haven't yet covered the below ...
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1answer
148 views

Are prime ideals always comaximal?

This is easy to see in the ring of integers. In fact, the ideals don't even have to be prime. It's enough to be coprime. Then their GCD is 1, so 1 can be written as a linear combination of the ...
2
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1answer
40 views

showing $\psi: R\to \mathbb C$ is ring isomorphism.

Below is an example from I.N. Herstein: Let $R=\Bigg\{\left( \begin{array}{ccc} a & b \\ -b & a \end{array} \right)\Bigg|a,b\in \mathbb R\Bigg\}$ and let $\mathbb C$ be the field of ...
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1answer
61 views

Prove that $S$ is a subring of $\mathbb{Z}_{28}$

Question: $S=\{0,4,8,12,16,20,24\}.$ Prove that $S$ is a subring of $\mathbb{Z}_{28}$ Confusion 1: This might be a dumb question, but when we refer to $[4]$ in $S$, for example, is that the congruent ...
4
votes
4answers
248 views

When are cancellations allowed in ring?

During the lecture my professored mentioned something like "cancellation is perfectly fine in a ring when dealing with addition, but not with multiplication!". The example he gave was that, in ...