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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21 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 $\mathbb{Q}[x,y]/(x,y)$ ...
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0answers
7 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, $1$ ...
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1answer
25 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
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1answer
16 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, ...
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1answer
66 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 ...
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0answers
29 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. ...
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0answers
12 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?
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1answer
13 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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0answers
23 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
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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 ...
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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
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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 ...
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3answers
54 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
38 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
26 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 ...
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0answers
42 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
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1answer
27 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
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3answers
124 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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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 ...
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1answer
50 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. ...
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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 ...
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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 ...
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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
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1answer
29 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
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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\} $.
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2answers
22 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 ...
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1answer
39 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
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 ...
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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 ...
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1answer
28 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 ...
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4answers
216 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 ...
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2answers
96 views

If a module is nonzero, then a localization module is nonzero

Let $R$ be a commutative ring, when $\mathfrak p$ is a prime ideal, there is the localization $M_{\mathfrak p}:=S^{-1}M$, where $S=R\setminus\mathfrak p$. Show: If $M$ is a nonzero $R$-module, ...
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2answers
30 views

A concrete example of a unital noncommutative ring without maximal two-sided ideals

Whenever I say ideal in this question I'm talking about two sided-ideals. Does there exist a concrete example of a non-commutative ring with $1$ without maximal ideals? We know that if $R$ is a ...
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2answers
40 views

If $R$ is a local ring, is $R[[x]]$ (the ring of formal power series) also a local ring?

So, I was trying to find a counter-example that shows not every local ring's lattice of ideals is a chain. I think $F[[x_1,\cdots,x_n]]$ is a good counter-example but I'm not able to show that ...
3
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1answer
31 views

Integral domain ${\mathbf Z}[\alpha] = \{a + b\alpha\ |\ a,\ b \in {\mathbf Z}\}$

Exercise: Show that the smallest subdomain of complex numbers containing the element $\alpha=\frac{\sqrt{5} - 1}{2}$ is ${\mathbf Z}[\alpha] = \{a + b\alpha\ |\ a,\ b \in {\mathbf Z}\}$. I thought I ...
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1answer
27 views

A property about quasi-primary modules

It is a fact that any discrete valuation domain $R$ has the property "P" that any proper submodule $N$ of any $R$-module $M$ is quasi-primary, in the sense that $\operatorname{rad}(N:M)$ is a prime ...
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1answer
30 views

Equivalent definitions of an algebra over a ring

I have seen two different definitions from several sources of an algebra over a ring. From Wikipedia: Let R be a commutative ring. An R-algebra is an R-Module A together with a binary operation ...
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1answer
39 views

What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$?

Question: What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$? My proof: (which I doubt whether its correct or not since it doesn't use the hint in the book) $[5^{2000}]=([5])^{2000}$ Since $5 \equiv ...
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0answers
48 views
+50

Basis of the ring $B=End_R(R^{(\mathbb N)})$

Let $B=End_R(R^{(\mathbb N)})$. Define $u,v \in B$ as $$u(e_{2_{i+1}})=0, u(e_{2_i})=e_i$$$$v(e_{2_{i+1}})=e_i,v(e_{2_i})=0$$ Prove that $\{u,v\}$ is a basis of $B$ as a $B$-module. I've already ...
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1answer
23 views

ring morphism from a group ring to another ring

I've read that if $S$ is a commutative ring, then $Hom_R(R[G],S)=Hom_R(R,S)\times Hom_{Gr}(G,\mathcal U(S))$. I've tried to show this equality but I couldn't. If $\phi: R[G] \to S$ is a ring ...
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1answer
32 views

Graded ring localization. Why is this function bijective? [duplicate]

From Hartshorne, Chapter II.2, Proposition 2.5(b). If $R$ is a graded ring and $\mathfrak a$ is a homogenous ideal, then the function defined as $$\phi(\mathfrak a) = \mathfrak aR_f\cap R_{(f)}$$ ...
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2answers
21 views

How to show that ideal is prime in $\mathbb{R}[x,y,z]$ modulo some other ideal

Let $R:=\mathbb{R}[x,y,z]$ and $g:=x^2+y^2-z^2\in R$. I would like to know how to show that $(x,y-z)/(g)$ is a prime ideal in $R/(g)$, and whether it is maximal or not. Thanks for the help!
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1answer
20 views

Polynomial rings of two variables

Prove that $(x,y)$ is not a principal ideal in $\mathbb{Q}[x,y]$. Here what is the definition of $(x,y)$? I don't know how to start the solution since I don't know the meaning of $(x,y)$.
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2answers
42 views

Noether normalization for $k[x]_{x}$

According to the Noether normalization theorem, there exists a $k[t]$ where $t$ is an indeterminate and $k[t]\subseteq k[x]_{x}$ is a $k$-algebra extension so that $k[x]_{x}$ is a finitely generated ...
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2answers
75 views

Is this ring a well known ring and if so how is it called?

I just had this thought when I was thinking how I was introduced to the concept of number in primary school and I came upon the conclusion that the numbers we were taught to manipulate (adding, ...
1
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1answer
21 views

Question on Wedderburn components of $\mathbb C[G]$

$G$ is a finite group. Wedderburn's theorem says that $\mathbb{C}G\cong R_1 \times\cdot \cdot \cdot R_r$ as rings where $R_i=M_{n_i}(D^i)$ is the ring of $n_i\times n_i$ matrices over a division ring ...
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2answers
15 views

Quick way to determine existence of integral root of a polynomial in one variable

Suppose $p(x) \in \mathbb{Z}[x]$ and if there exist $b \in \mathbb{Z}$ s.t. $p(b)=0$, then $x-b|p(x)$. The other technique can be to put all $b \in \mathbb{Z}$. But this require to check every $b \in ...
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0answers
17 views

direct sum of modules and generator subset

I am trying to solve the following problem: Let $(M_i)_{i \in I}$ be an infinite family of non zero modules and $S$ a system of generators of $\bigoplus_{i \in I}M_i$. Prove that the cardinal of $S$ ...
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0answers
45 views

Division Rings and trivial ideals

I'm stuck in the following exercise, I guess this is easy but I would appreciate someone's tip.... I have to prove the following: If $R$ is a right simple ring (the unique right ideals are $R$ and ...
1
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1answer
24 views

Characterize semisimple rings with a unique maximal ideal

Problem Characterize the semisimple rings $R$ that contain a unique maximal ideal. I am not so sure what to do here. I know that a ring $R$ is semisimple if and only if all $R$-modules are ...