Tagged Questions

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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1
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1answer
22 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.
1
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1answer
13 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 ...
-3
votes
0answers
25 views

Which of the following are correct? [on hold]

Let $R$ be the ring obtained by the taking the quotient of $(\mathbb{Z}/6\mathbb{Z})[X]$ by the principal ideal $\langle2X+4\rangle$. Then A. $R$ has infinitely many elements. B. $R$ is field. C $5$ ...
2
votes
2answers
27 views

R commutative Ring, then $R^{\times}\subset (R[X])^{\times}$

Let R commutative Ring. Also there is a $a\in R$ such that $a^n=0$ then $R^{\times}\subset(R[X])^{\times}$ (strictly "subset of") So, there is an element in $(R[X])^{\times}$ which is not contained in ...
2
votes
1answer
35 views

Associates in a non-integral domain

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 ...
0
votes
0answers
11 views

Probability of finding roots to a multivariate polynomial [on hold]

Let there exist a $n$-variate $1$-degree polynomial ring. Suppose there is a finite set $S$ as a subset of the field over which the polynomial ring is defined. What is the probability that elements ...
2
votes
2answers
34 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 ...
0
votes
3answers
25 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.) $\mathbb Z[x]$
-1
votes
1answer
15 views

$\rm{End}_\mathbb Z(\mathbb Z_n)$ [on hold]

Let $\mathbb Z_n$ be $\mathbb Z$-module. I am looking for $\rm{End}$$_\mathbb Z(\mathbb Z_n)$ where by $\rm{End}$$_\mathbb Z$ I mean the ring of all $\mathbb Z$-module endomorphisms.
0
votes
1answer
7 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 ...
0
votes
1answer
37 views

homology and cohomology with coefficients of ring and field [on hold]

(1). Let $R$ be a ring. Let $X$ be a topological space. Then $H^n(X;R)$ is a module over $R$. Also $H_n(X;R)$ is a module over $R$.Is this statement correct? (2). Let $F$ be a field. Let $X$ be a ...
0
votes
2answers
29 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 ...
0
votes
1answer
11 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 ...
1
vote
1answer
26 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 ...
2
votes
0answers
28 views

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

How can I write $210\in\mathbb Z[\sqrt{-1}]$ as a product of prime elements?
1
vote
1answer
30 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
35 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 ...
2
votes
1answer
29 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 ...
-1
votes
0answers
21 views

If $R$ is a ring, $S$ is a subring, and $I$ is an ideal of $R$, then $I\cup S$ is an ideal of $S$

Proof or Counterexample: If $R$ is a ring, $S$ is a subring, and $I$ is an ideal of $R$, then $I\cup S$ is an ideal of $S$. Just before this I encountered a similar problem with $I\cap S$ and was ...
1
vote
0answers
31 views

Show that $deg(f\cdot g)=n+m$

I started learning about rings and I was asked to proof some claims. I don't understand how I may prove the last one. I have proven that if $f$ and $g$ are polynomials over some ring of polynomials, ...
-3
votes
5answers
42 views

What is an example of a commutative ring with a non-zero element [on hold]

True or false? For every element r in a ring R, if r≠1, then 1-r is invertible. i have tried using the inverse relation and was not able to find the answer may i know how we can verify this .I think ...
1
vote
2answers
26 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
0answers
10 views

Classifying Unital commutative Rings of order 8

Classify unital commutative rings of order 8. Attempt:In a unital ring $R$ of order $8$ the additive order of $1$ can only be $2,4$ or $8$. In the third case when additive order of $1$ is 8, $1$ ...
0
votes
1answer
33 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
17 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, ...
2
votes
1answer
75 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
35 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
votes
0answers
17 views

Irruducible $R$-modules of a ring [on hold]

Suppose $R$ is a ring and $M$ is an $R-$module. If $M$ is irreducible, then what is the meaning of it?
0
votes
1answer
14 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 ...
1
vote
1answer
33 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?
1
vote
1answer
36 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
votes
2answers
30 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
vote
1answer
22 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 ...
1
vote
3answers
56 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, ...
0
votes
1answer
43 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
-2
votes
0answers
32 views

Let R be a ring. Suppose that there exists an element r∈R with r^n = 0, for some n ≥ 1. [on hold]

Let R be a ring. Suppose that there exists an element r∈R with r^n = 0, for some n ≥ 1. Prove that 1 - r is invertible. (Hint: to get some ideas, think of the equality 1/(1-t)=∑_(k=0)^∞▒t^k, valid for ...
0
votes
0answers
46 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 ...
1
vote
1answer
28 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 ...
3
votes
3answers
125 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 ...
0
votes
2answers
36 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 ...
1
vote
0answers
57 views

True or false? For every element $r$ in a ring $R$, if $r\neq 1$, then $1 − r$ is invertible. [on hold]

True or false? For every element $r$ in a ring $R$, if $r\neq 1$, then $1 − r$ is invertible. ...
1
vote
1answer
40 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 ...
0
votes
1answer
34 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 ...
0
votes
2answers
50 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 ...
1
vote
1answer
32 views

Prove or disprove : $a_0+a_1 x+…+a_n x^n\in R[x]$ is nilpotent iff $a_i$ is nilpotent $\forall i$.

1) Prove or disprove : $a_0+a_1 x+...+a_n x^n\in R[x]$ is nilpotent iff $a_i$ is nilpotent $\forall i$. 2) Prove or disprove : $a_0+a_1 x+...+a_n x^n\in R[x]$ is a unit iff $a_i$ is nilpotent ...
1
vote
1answer
32 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\} $.
1
vote
2answers
23 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
votes
1answer
41 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 ...
1
vote
1answer
21 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 ...
1
vote
1answer
18 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 ...