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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4
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1answer
67 views

Does any (noetherian) integral domain have a “UFD closure”?

Let $R$ be a (possibly noetherian if that helps) commutative unital integral domain. Does there exist a UFD $\overline{R}$ such that $R$ embeds in $\overline{R}$ (via some map $\psi$) and such that ...
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0answers
28 views

Homomorphic images of $\mathbb{Z}[x]$

How to prove that any finite field is a quotient ring of $\mathbb{Z}[x]$ ? I am not sure whether this result is true or false. Any hint will be appreciated. Thanks in Advance.
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0answers
28 views

number of elements in the quotient ring $\frac{\mathbb{Z}_n[x]}{<ax+b>}$

How to find the number of elements in the quotient ring $\frac{\mathbb{Z}_n[x]}{<ax+b>}$ where n is a composite number. In particular what is the number of elements in the ring ...
1
vote
1answer
49 views

Example of a semi-simple $\mathbb{R}$ algebra

Let $[n]:=\{1,....,n\}$ and define the $2^n$-dimensional $\mathbb{R}$-algebra $C_n$ with basis $e_I$, $I \subset [n]$, such that $e_\emptyset = 1, e_ie_j = -e_je_i$ for $i \not =j, e_j^2 = 1 $ and ...
0
votes
1answer
13 views

Explaining a. Q(R,T) has unity even if R does not and b. In Q(R,T) every nonzero element of T is a unit

So I've been having trouble understanding this proof for quite a while now. I understand how the field of quotients is formed but not so much of why my professor's answers use this tactic for this ...
3
votes
3answers
94 views

Show that quotient rings are not isomorphic

I've been given a homework problem that requires me to show that the rings $\mathbb{C}[x,y]/(y - x^2)$ and $\mathbb{C}[x,y]/(xy-1)$ are not isomorphic. This is my attempt at a solution: For ...
1
vote
2answers
36 views

Localization Question: $\frac{a}{b}\in\left(\mathbb{Z}_{(p)}\right)^{\times}\iff\frac{b}{a}\in\mathbb{Z}_{(p)}$

Questions: $\rm\color{#c00}{(1)}$ Is the $[\Longrightarrow]$ implication of $$ \frac{a}{b}\in\left(\mathbb{Z}_{(p)}\right)^{\times}\iff\frac{b}{a}\in\mathbb{Z}_{(p)} $$ obvious? ...
1
vote
1answer
29 views

showing Module is simple

Given the following: let $C \subset \mathbb{H}$ be a subring of the real quarternion algebra such that it contains the center of $\mathbb{H}$ = $Z(\mathbb{H})$ Also C $\cong \mathbb{C}$ Then let R ...
2
votes
1answer
43 views

Jacobson radical of a certain ring of matrices

Given a Matrix $A \subset M_4(\mathbb{C})$ be the $\mathbb{C}$-subalgebra consisting elements in the form \begin{pmatrix} * & * & * & *\\ * & * & * &*\\ 0 & 0 & ...
3
votes
2answers
57 views

Jacobson radicals of $R$ and $R/I$ where $I$ is a nilpotent ideal.

Out of interest If i have the map $\phi: R \longrightarrow R/I $ where $R$ is a ring and $I$ is a nilpotent ideal ? then would i be right in saying that if i were to apply this map to the jacobson ...
2
votes
0answers
33 views

To find all Ring homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ [duplicate]

How to find all Ring homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ (with the usual ring structure ) ?
3
votes
0answers
39 views

Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
0
votes
2answers
47 views

Assume we have $\mathbb{Z}_{p}[x]$ with $p$ being a prime. Prove that $x^{p-1}-1=(x-1)(x-2)…(x-(p-1))$

I know how this formula works and it is quite interesting actually but how would you prove this relationship? Through induction (seems difficult since there's no equation for prime numbers), but I'm ...
0
votes
1answer
26 views

Verification of proof that for a,b in ring R, assuming ab is a zero divisor at least one of a and b is zero divisor

I'm not so sure if this is correct but here's what I have so far: ab is a zero divisor iff there is a c$\neq$0 s.t. (ab)c=c(ab)=0 given ab$\neq$0 and c$\neq$0. Then we have ...
2
votes
0answers
45 views

Question on an $\mathbb{R} $-algebra

Define $[n] = \{1,\ldots, n\} $, where $n \in \mathbb{N}$ and define the $2^n$- dimensional $\mathbb{R}$-algebra $C_n$ as follows: Notation: Basis is $e_I$, where $I \subset \mathbb{N}$ and let ...
1
vote
2answers
40 views

Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$

Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$. I'm not entirely sure how to tackle the "infinitely many elements ...
0
votes
2answers
49 views

Verification of proof that if $R$ is a commutative ring, $a$ is a unit and $b^2=0$ then $a+b$ is also a unit

Here's what I have so far and I would like to know if I am right or if my proof needs to be edited: Since $a$ is a unit it means $a1=a$, with $1$ being the unity element We know $b^2=0$ and this ...
0
votes
1answer
29 views

How many polynomials in $Z_{p}[x]$ have degree n or less?

For your reference, $Z_{p}[x]$ refers to the set of all polynomials with coefficients integer mod p. To me it seems like this and the degree (power) of the two polynomials are unrelated. What ...
1
vote
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
vote
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
vote
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
votes
3answers
61 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
votes
2answers
31 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
votes
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
votes
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
votes
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 ?
0
votes
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.
3
votes
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
votes
2answers
38 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
vote
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
votes
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
67 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
votes
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 ...
1
vote
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
39 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
vote
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
votes
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
60 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
votes
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
votes
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
vote
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
vote
1answer
92 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
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 ...