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
77 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
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1answer
84 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
votes
2answers
94 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
vote
2answers
85 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
61 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
325 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
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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
votes
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 $...
1
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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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votes
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 $a\in\...
0
votes
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
vote
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 $A$...
1
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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, ...
1
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1answer
600 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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votes
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 \mathbb{Z}_{/<n>} \\...
1
vote
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 ...
4
votes
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 ...
0
votes
2answers
151 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
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 ...
0
votes
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 ...
0
votes
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
votes
1answer
115 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: $a_0+a_1 x+\cdots+a_n x^n\in R[x]$ ...
1
vote
1answer
64 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
149 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
108 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
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1answer
150 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
votes
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 ...
1
vote
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 $\...
0
votes
2answers
234 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
123 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 ...
0
votes
2answers
164 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 $F[[x_1,...
3
votes
1answer
57 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 ...
1
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1answer
45 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 ...
3
votes
1answer
205 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 [·,...
1
vote
1answer
51 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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1answer
91 views

Basis of the ring $B=\operatorname{End}_R\left(R^{(\mathbb N)}\right)$

Let $B=\operatorname{End}_R\left(R^{(\mathbb N)}\right)$. 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 ...
1
vote
1answer
43 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 morphism,...
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1answer
57 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
46 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!
0
votes
1answer
31 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
128 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 $...
8
votes
2answers
95 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
59 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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votes
2answers
30 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 ...
1
vote
1answer
49 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 ...
1
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1answer
42 views

Prove or disprove statements about modules

I am trying to determine if the following statements are true or false (i) There are free modules with non zero elements $x$ such that $\{x\}$ is linearly dependent. (ii) There are non free modules ...
1
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1answer
67 views

Counterexample of “the product of open subsets is open in a topological ring”?

Given a topological ring $R$ and $U,V$ open subsets, we can show that $U+V$ is an open subset due to the fact that $x\mapsto x+y$ is a homeomorphism for every $y \in R$. Since, in general, $R$ is not ...