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
36 views

Verification of Proof: Let $R$ be a ring of unity and $a \in R$ satisfy $a^2=1$. $S=\{ara \mid r \in R\}$ is a subring

Here's what I got. The three conditions we have to prove are: $0$ is in $S$: Let $r=0$ and this implies $a0a=0a=0$ which is in $S$ $(a-b)$ is in $S$ for all $a,b \in S$: Let $a, b \in S$. this ...
1
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2answers
16 views

$x$ in intersection of maximal ideals implies $1-x$ is a unit

Let $R$ be a commutative ring, we define $J:=\bigcap_{\mathcal M \space \text{maximal}}\mathcal M$. Let $x \in J$, prove the following $(1-x) \in \mathcal U(R)$ If $x^2=x$ then $x=0$ For the ...
1
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2answers
38 views

Computing the center of a ring

if i have the following ring $R = \mathbb{H} \otimes _\mathbb{R} M_2(\mathbb{C}) $ then how would i find the center $Z(R)$? Also is this ring simple, i am sure it is but am struggling to show that ...
2
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3answers
59 views

inverses in $R/I$ where $I$ is a nilpotent ideal

Given an element $x \in R$ where R is a ring $I$ is a nilpotent ideal of $R$, i am trying to find inverses in the quotient R/I and thought about things in the general case, what would determine the ...
2
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2answers
28 views

A question about von Neumann regular rings and their ideals

Suppose $R$ is a von Neumann regular commutative ring with a unit. Prove that every principal ideal $I$ is generated by an idempotent element and for every principal ideal $I$, there exists a ...
0
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1answer
48 views

Discrete Valuation Rings - Atiyah & MacDonald

The following is claimed (without much proof) during the the proof of Prop 9.2 in Atiyah & MacDonald. Saurabh commented below giving the proof that was probably intended by A&M (thank you!). I ...
0
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1answer
70 views

When is the quotient ring of a multivariable polynomial ring over a field an integral domain?

When is the quotient ring of a multivariable polynomial ring over a field an integral domain? I am actually trying to show that a monomial ideal is prime by showing the corresponding quotient ...
0
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1answer
34 views

Questions concerning $\mathbb Z_3[x]/(x^3+2x-1)$

Is the automorphism group of $\mathbb Z_3[x]/(x^3+2x-1)$ cyclic ? Is $\mathbb Z_3[x]/(x^3+2x-1)$ separable ?
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1answer
24 views

Should Ext-quiver be a full sub-quiver of its AR-quiver for a basic hereditary algebra A over algebraic closed field K?

For a basic hereditary algebra A over algebraic closed field K, prove its Ext-quiver $\Gamma_{A}$ is a full sub-quiver of its AR-quiver $\Delta_{A}$. I have no clue for this.
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0answers
36 views

Henselization of the ring of polynomials

I am trying to understand example of Henselization from wiki. http://en.wikipedia.org/wiki/Henselian_ring#Henselization It says that Henselization of the ring of polynomials localized at point $(0, ...
2
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2answers
37 views

Prove every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.)

Let $A$ be a commutative ring with unity. Prove: Proof every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.) In the question before ...
1
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1answer
25 views

Factor $55 - 88 \sqrt{-2}$ as a product of primes in $\mathbb{Z}[\sqrt{-2}]$

To solve this problem, I let $K = \mathbb{Q}(\sqrt{-2})$, and I thought to take the norm $$N(55 - 88 \sqrt{-2}) = 55^2 + 2 \cdot 88^2 = 18513 = 3^2\cdot11^2 \cdot 17$$ If $a \in \mathbb{Z}[\sqrt{-2}]$ ...
0
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0answers
27 views

Wedderburn-Artin decomposition of the group algebra of the cyclic group of 3 elements over different fields

Determine the Wedderburn-Artin decomposition of the group algebra $FG$, where $G$ is the $3$-element group $G=\{g,g²,g³=1\}$ and $F$ is the field (i)$\mathbb{C}$ (ii)$\mathbb{R}$ (iii)$\mathbb{Q}$ ...
3
votes
1answer
66 views

In a Noetherian integral domain, a principal prime ideal can't have proper non-zero prime ideals

Let $R$ be an integral domain and Noetherian. Let $P \subset R$ be a non zero prime ideal. Prove that if $P$ is principal then there is no $Q$ prime ideal such that $0 \subsetneq Q \subsetneq P$. ...
0
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0answers
6 views

for every n exists ideal in $\mathbb Z [x,y]$ which is generated by exactly n elements.

I have tried to prove that in $\mathbb Z [x,y]$ - the polynomial ring with two variables and integer coefficients, for every $n\in\mathbb N$ exists an ideal which is generated by $n$ elements and ...
0
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1answer
50 views

$\mathbb R[X]/\langle X^4-1\rangle \cong \mathbb R \times \mathbb R \times \mathbb C$

I am trying to prove the isomorphism $\mathbb R[X]/\langle X^4-1\rangle \cong \mathbb R \times \mathbb R \times \mathbb C$. I will write what I did so you can help me from there. First notice that ...
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3answers
73 views

Calculate the dimension of the field extension $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}]$

I've though that $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}] = [\mathbb{Q}[ \sqrt2] : \mathbb{Q}].[ \mathbb{Q}[\sqrt2, \sqrt3]:\mathbb{Q}[ \sqrt2] ] $ And I know how to prove $[\mathbb{Q}[ \sqrt2] : ...
2
votes
1answer
37 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
23 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 ...
1
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1answer
48 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
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1answer
47 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 ...
2
votes
1answer
33 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., $$ ...
3
votes
1answer
59 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 ...
4
votes
1answer
28 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
31 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
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1answer
25 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!
2
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2answers
72 views

Homomorphism $\mathbb{R}\to \mathbb{Q}$ [closed]

Is possible to define non-trivial homomorphisms from $\mathbb{R}\to \mathbb{Q}$?
2
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2answers
32 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 ?
1
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1answer
29 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
48 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
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1answer
91 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
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2answers
74 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 ...
0
votes
1answer
32 views

Any ring of prime order commutative ?

Is any ring of prime order commutative ?
4
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3answers
232 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 ...
0
votes
1answer
28 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
34 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 of ...
0
votes
1answer
20 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
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1answer
35 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 ...
0
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0answers
32 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
11 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 ...
3
votes
1answer
65 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 ...
2
votes
2answers
25 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
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1answer
152 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 ...
2
votes
1answer
51 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
53 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 ...
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2answers
69 views

Ring with nested prime ideals [closed]

If $n>1$ is there a (commutative with identity) ring with Krull dimension $n$ and only $n+1$ prime ideals?
2
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1answer
25 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
42 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 ...
0
votes
2answers
51 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
17 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 ...