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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2
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1answer
71 views

Begginer doubt in Ring of p-adic integers

I am studying $p$-adic Rings and let me explain my understanding and doubt here. As I understood, Let $p$ be a rational prime and $Z$ denotes ring of integers, then form cartesian product $$P=Z/pZ \...
2
votes
1answer
31 views

Show that a ring is simple

In the ring $R = \mathbb{H} \otimes_{\mathbb{R}} M_{2}(\mathbb{C})$ I have computed the center as $Z(R)= \mathbb{C}$. I am however struggling to show that $R$ is a simple ring and consequently find ...
2
votes
1answer
102 views

Prove that a ring $R$ with no non-trivial right ideals and $aR=0$ has $|R|=p$ prime

Let $R$ be a ring such that $R$ has no non-trivial right ideals. If there exists a nonzero element $a \in R$ with $aR=0$, prove that $|R|= p$ where $p$ is prime.
2
votes
1answer
184 views

Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$

Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$. I've always done matrix arithmetic in a field $\mathbb F$ and that is what every ...
2
votes
1answer
60 views

cohomology monomorphism between grassmannians and product of projective spaces

Let $S^1\times\cdots \times S^1( n\text{ times })=\prod_n U(1)\to U(n)$ be the inclusion. This induces a map between classifying spaces $$ \prod_nBS^1\to BU(n).$$ i.e., $$ (\mathbb{C}P^\infty)^{\...
4
votes
1answer
234 views

Two elements in a non-integral domain which are not associates but generate the same ideal

Let $\mathbb{K}$ be a field. Let $R$ be the quotient ring $\mathbb{K}[x,y]/(xy^{2})$. Let $\bar{x}$ be the class of $x$ in $R$ (i.o.w. $\bar{x}=x+(xy^{2}))$. Prove that $\bar{x}$ and $\bar{x}+\bar{x}...
4
votes
1answer
72 views

Irreducible radical ideals are prime

Assume $R$ is a commutative ring and $I$ is a nonzero proper ideal of $R$ satisfying: $(1)$ If $I_1$ and $I_2$ are ideals such that $I = I_1 \cap I_2$, then $I = I_1$ or $I = I_2$; $(2)$ If $a^n \...
2
votes
0answers
34 views

Can we build infinite products in $k[[X]]$?

Let $P \in k[[X]]$, where $k[[X]]$ denotes the ring of formal power series over the field $k$. Is well defined $$\prod_{n\in \mathbb{N}}P$$ (i.e. the power to infinity of $P$?) By looking at the ...
0
votes
1answer
36 views

$R=\mathbb Z[\sqrt3]. x=2-sqrt3, then {x^n:n is an integer} is an infinite set of distinct values.

Let $R=\mathbb Z[\sqrt3]$ I would like to show that when $x=2-\sqrt3$, then $\{x^n:n \in \mathbb Z\}$ is an infinite set of distinct values. How should I do this? Thank you!
1
vote
2answers
35 views

$R$ is a division ring and $a \in R$ then is $N(a):=\{x\in R : xa=ax \}$ a division ring? [closed]

If $R$ is a division ring and $a \in R$ then is $N(a):=\{x\in R : xa=ax \}$ a division ring ?
2
votes
1answer
88 views

Jacobson radical of a commutative ring

Let $R$ be a commutative ring, $I$ be a minimal ideal of $R$. Prove that for all $y$ belong to $Rad(R)$, $yI=0$. ($Rad(R)$ denotes the Jacobson radical of $R$) $Rad(R)$ equals the intersection of all ...
0
votes
1answer
66 views

Localization of Rings: Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$

Let $R$ be a ring, $f \in R$, and $X$ a variable. Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$. I am a beginner in algebra and I am reading a textbook in commutative algebra. What I do not ...
1
vote
1answer
148 views

Group and Ring Homomorphisms between $\mathbb{Z}_p$ and $\mathbb{Z}_{2p}$

Let $p$ be prime with $p > 2$. (a) Determine the number of group homomorphisms between $\mathbb{Z}_p$ and $\mathbb{Z}_{2p}$ (b) Determine the number of ring homomorphisms between $\mathbb{Z}_p$ ...
0
votes
2answers
82 views

Prove $S^{-1}\mathbb{Z_6}\simeq \mathbb{Z}_3$

Prove $S^{-1}\mathbb{Z_6}\simeq \mathbb{Z}_3$ when $S=\{\overline{1},\overline{2},\overline{4}\}$. Note: $S^{-1}\mathbb{Z_6}= \frac{S\times \mathbb{Z}_6}{\sim }$ where $(x,y)\sim (u,v) \iff \exists ...
1
vote
1answer
293 views

Any ring of prime order commutative ?

Is any ring of prime order commutative ?
4
votes
3answers
241 views

How to show distributivity in a ring, and what is wrong with my algebra?

I am trying to show the following is a commutative ring with unity, however I am encountering a problem. First, addition and multiplication are defined as: $$a \oplus b=a+b-1$$$$a \odot b=ab-(a+b)...
0
votes
1answer
29 views

Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$

Prove that: $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$ We just learned about the characteristic/minimal polynomial and diagonalization but I am not sure if it has ...
2
votes
1answer
61 views

Example of non-commutative ring with exactly 2014 two sided-proper ideals.

Find a non-commutative ring with exactly 2014 two sided-proper ideals. Find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have thought ...
0
votes
1answer
32 views

Minimal right ideals

Let $I$ be a minimal right ideal of a ring $R$ with $1$. If $r\in R$, could we say that $rI$ is zero or a minimal right ideal? I assumed a right ideal $J$ in $rI$ and intersecting it with $I$ got a ...
1
vote
1answer
90 views

homomorphism between cohomology induced by the multiplication of an H-space

Define the product on $\mathbb{C}P^\infty$ in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow \mathbb{C}P^\infty,...
0
votes
0answers
41 views

Doubt related to quotients (group or ring)

I was reading some notes about ring theory and modules and I've encountered with the following isomorphism: $\mathbb (R[X]/ \langle x^3-1\rangle)/ \langle x-1\rangle \cong \mathbb R[X]/ \langle x-1 \...
0
votes
1answer
24 views

common factors of multilinear polynomial

Say $F,G\in\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ are two multilinear polynomial. If $F$ and $G$ vanish at a common set of coordinantes $(a_{i1},a_{i2},\dots,a_{in-1},a_{in})\in\Bbb R^n$ for $i=1,\dots,t$...
5
votes
1answer
215 views

If $n\mid m$ prove that the canonical surjection $\pi: \mathbb Z_m \rightarrow \mathbb Z_n$ is also surjective on units

Not sure if this is the right proof (i found it online): Since $n\mid m$, if we factor $m = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, then $n = p_1^{\beta_1}p_2^{\beta_2}\cdots p_k^{\...
2
votes
2answers
41 views

Computing the inverse of an element in $\mathbb{Z}_5[x] / \langle x^2+x+2\rangle$

How does one calculate the inverse of $(2x+3)+I$ in $\mathbb{Z}_5[x] / \langle x^2+x+2\rangle$? Give me some hint to solve this problem. Thanks in advance.
5
votes
1answer
185 views

algebra with topology homework problem

Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have: $$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists \,U ...
2
votes
1answer
110 views

Simple questions about the Jacobson Radical

Questions: [See below] $\rm\color{#c00}{(1)}$ What structure has $R/I$ when $I$ is a maximal left ideal? I know that if $I$ is a bilateral ideal then $R/I$ is a ring and, moreover, if $I$ is ...
2
votes
1answer
78 views

Counterexample for the infinitely many primes between two primes in a Noetherian ring

Consider the following Proposition: Proposition: Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many ...
0
votes
1answer
36 views

Multiplicative group of a field contains maximal n-1 elements with order n

Let $F$ be a field and $n\in \mathbb N,n>1$. I want to show that the multiplicative group $K$\ $\{0\}$ contains maximal $n-1$ elements with order $n$. I actually don't have any ideas how to solve ...
1
vote
1answer
65 views

A finite-dimensional $K$-algebra?

Let $A$ be a commutative noetherian domain of characteristic $p>0$ and $K$ be its quotient field. Let $G$ be any finite group whose order is divisible by $p$. Is $KG$ a finite-dimensional $K$-...
0
votes
2answers
79 views

Looking for help, Aluffi Exercise 5.13, Chapter 6: characterization of PIDs

I quote: "Let $M$ be a finitely generated module over an integral domain $R$. Prove that if $R$ is a PID, then $M$ is torsion-free if and only if it is free. Prove that this property characterizes ...
0
votes
1answer
33 views

$M_n(D)$ is left and right-simple?

Is it true that if $D$ is a division ring and $n\in\mathbb{Z}_{\geq1}$, then the only left and right ideals of the ring $M_n(D)$ are the trivial ones? I know that $M_n(D)$ is simple, and the proof ...
0
votes
1answer
154 views

Steps to construct the Field of fractions of Gaussian Integers $\mathbb{Z}[i]$ [duplicate]

i don't know how to construct such field $\mathbb{Q[i]} $ from $\mathbb{Z[i]}$. I know the following: $(a+bi,c+di)\sim (m+ni,r+si)$ iff $(a+bi)(r+si)=(c+di)(m+ni)$ is the equivalence relation and if $\...
1
vote
1answer
118 views

Atiyah & MacDonald on local Noetherian and Artinian rings - sanity check.

In the chapter on Artinian rings in "Introduction to Commutative Algebra" by Atiyah and MacDonald, we have: Proposition 8.6. Let $(A,\mathfrak{m})$ be a local Noetherian ring. Then exactly one of the ...
1
vote
1answer
56 views

Correspondence between ideals of $R$ and $D^{-1}R$

Let $R$ be an integral domain, and $D\subset R$ be a multiplicatively closed subset such that $1\in D$ and $0\not\in D$ . Prove/disprove that there is a one-to-one correspondence between the ideals of ...
2
votes
2answers
92 views

Trouble on Aluffi's Exercise 5.5, Chapter 6: finitely generated projective module over a local ring is free

The overall goal of the problem is the following: Let $R$ be a local commutative ring (so that there is a unique maximal ideal $\mathfrak{m}$) and let $M$ be a finitely generated $R$-module. Assume ...
2
votes
1answer
59 views

Suppose that $R$ is a commutative ring with unity, $a \in R$, and $\varphi(r) = ar$ defines a ring homomorphism. Prove that $a$ is idempotent.

Suppose that $R$ is a commutative ring with unity, $a \in R$, and $\varphi(r) = ar$ defines a ring homomorphism $\varphi$. Prove that $a$ is idempotent, i.e. that $a = a^{2}$. This is exercise 15 ...
1
vote
1answer
84 views

Is $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\bigotimes_{\mathbb{C}}\mathbb{C}$? [duplicate]

I'm trying to see if for several cases changing the ring in a tensor product affects the result or doesn't. Now I'm trying to prove $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\...
1
vote
1answer
36 views

Proving that this mapping is one to one

Let $Q$ be the field of quotients of the Gaussian integers (integer complex numbers) and let $R$ be the the set of all complex numbers of the form $a +bi$ such that both $a,b$ are rationals I have to ...
0
votes
2answers
72 views

Concerning Ideals and invertible elements in a commutative ring

Here is the problem that I have: Let $R$ be a commutative ring with unity and let $I$ be an ideal in $R$. Prove that $I=R$ if and only if $I$ contains some invertible element of the ring $R$. Here ...
-1
votes
1answer
117 views

If $f$ is a unit in a polynomial ring then $a_0$ is unit and all other coeficients are nilpotent.

I'm trying to prove the converse of the following theorem. I think suggestion available at this website are mistaken or I didn't understand them correctly. Theorem. Let $R$ be a commutative ring with ...
0
votes
1answer
376 views

Quotient field of gaussian Integers

Let $D$ be the set of all gaussian integers in the from of $m+ni$ where $m,n \in Z$ Carry out the construction of the quotient field $Q$ for this integral domain.Show that this quotient field is ...
2
votes
1answer
49 views

Questions about ring of smooth functions

First of all, this is a homework problem. Let $C^{\infty}(\mathbb{R})$ denote the ring of smooth functions. Let $I_n$ denote the set of $f\in C^{\infty}(\mathbb{R})$ such that $$f^{(k)}(0)=0, \ 0 \...
2
votes
1answer
91 views

Why $F[x]/p(x)$ would contain $F$?

I am reading Abstract Algebra by Hungerford, and I am really confused about how we can extend a ring to a bigger ring. Here's what I got from the book: $F$ be a field and $p(x)$ be a nonconstant ...
1
vote
1answer
87 views

Proving the ring $\mathbb{Q}$[$\mathbb{Z}$] is not artinian

My proposed solution: For each $n \in \mathbb{N}$, $\mathbb{Q}$[$2^{n}\mathbb{Z}$] is an ideal of $\mathbb{Q}$[$\mathbb{Z}$] (I think) and so we have the following infinite descending chain of ideals:...
0
votes
2answers
71 views

Matrix rings and ideals

How would one go about checking if a given 2 x 2 matrix is an ideal. I am unclear as to what an ideal is and would like to know the steps in order to make the verification. Also, if it helps, I had ...
2
votes
1answer
274 views

$I$ is an ideal in $R$ implies that $I[x]$ is an ideal in $R[x]$.

Is the following statement right? If $I$ is an ideal in the ring $R$, then $I[x]$ is an ideal in the polynomial ring $R[x]$. If so, how can I prove it?
0
votes
1answer
50 views

Factoring isomorphism

I have $\mathbb Z[i\sqrt2] = ${$a+bi\sqrt2; a,b \in \mathbb Z, i^2=-1$} and $I =\{a+bi: a,b \in \mathbb Z, i^2=-1, 11\mid a+3b\}$. My task was to prove that $I$ is an ideal in $\mathbb Z[i\sqrt2]$ by ...
1
vote
1answer
52 views

Showing certaing Integral domain is not well ordered.

Let $\mathbb{Z}[\sqrt2]=\{ a+b\sqrt2 ~| a,b\in \mathbb{Z}\} $ be an integral domain. Let $p=\{ a+b\sqrt2 ~|a,b\in\mathbb{Z} ~~~ and ~~~ a+b\sqrt 2 ~~is~~a~positive~real~number \} $ Show that $\...
2
votes
2answers
441 views

Is that Ring a field?

Given a commutative Ring $R$ of ordered pairs $(x,y)$ of reals $x,y$ with addition and multiplication defined in the following way. $$(x,y) + (u,v) = (x+u,y+v)$$ $$(x,y).(u,v) = (xu-yv,xv + yu)$$ I ...
0
votes
3answers
33 views

Does the following Multiplication have nonzero divisors

If we define $(x,y).(u,v) = (xu - yv,xv + yu)$ do we have any non zero divisors for this meaning can we find non zero elements $(x,y)$ and $(u,v)$ such that $(x,y).(u,v) = (0,0)$ i tried to think of ...