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
34 views

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

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
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1answer
19 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
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1answer
8 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 ...
0
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1answer
38 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
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1answer
22 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 ...
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2answers
22 views

Trouble on Aluffi's Exercise 5.5, Chapter 6: direct summand over a local ring is a free module

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 ...
1
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1answer
34 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 ...
2
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1answer
24 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
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1answer
40 views

algebra of polynomials of non-commuting variables

In Hatcher's book algebraic topology page 496-497: the Steenrod algebra $\mathcal{A}_2$ is the algebra over $\mathbb{Z}_2$ formed by polynomials of non-commuting variables $Sq^1,Sq^2,\cdots$ modulo ...
1
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1answer
67 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 ...
2
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3answers
164 views

Commutative integral domain does not finitely generate its field of fractions

I want to prove that if we have a commutative integral domain $D$ with field of fractions $F\neq D$ then $F$ is not finitely generated as a $D$-module. (In this question it may be the case that ...
0
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1answer
34 views

Show that field of fraction of a commutative domain is an indecomposable module which is not finitely generated

I came across this problem and get stuck for quite sometime. Problem: Let $R$ be a commutative domain that is not a field. Let $F$ be its field of fractions. Show that $F$ is an indecomposable ...
9
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3answers
2k views

Characterizing units in polynomial rings

I am trying to prove a result, for which I have got one part, but I am not able to get the converse part. Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots ...
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2answers
30 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 ...
1
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1answer
58 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 ...
1
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1answer
23 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 ...
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2answers
14 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 ...
0
votes
1answer
36 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 ...
3
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1answer
79 views

Proof for Unique Factorization Domain

Prove that the quotient ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a unique factorization domain. I am trying to prove first it is a principal ideal domain. However I am really stuck on this problem
2
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1answer
67 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 ...
2
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1answer
23 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
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2answers
52 views

Integral dependence over rings is transitive

Let $A\subset B\subset C$ be commutative rings. Suppose $B$ is integral over $A$, and $C$ is integral over $B$. Then I want to show that $C$ is integral over $A$. To be integral means that for ...
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0answers
24 views

Factorring isomorphism

I have $\mathbb Z[i\sqrt2] = ${$a+bi\sqrt2; a,b \in \mathbb Z, i^2=-1$} and $I =${$a+bi; a,b \in \mathbb Z, i^2=-1, 11| a+3b$ }. My task was to prove that $I$ is an ideal in $\mathbb Z[i\sqrt2]$ by ...
0
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0answers
21 views

Ramification index multipicative

Let $R\subseteq R'\subseteq R''$ be Dedekind rings and P a non-zero prime ideal in $R''$ .I need to show that $e(P/R)=e(P/R')e(P\cap R'/R)$ where $e(P/R)$ is the ramification index of P in respect of ...
2
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1answer
19 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?
2
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1answer
17 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
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2answers
381 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 ...
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3answers
25 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 ...
0
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3answers
44 views

If $\gcd(a,b) = 1,$ then why is the set of invertible elements of $\mathbb Z_{ab}$ isomorphic to that of $\mathbb Z_a\times \mathbb Z_b$?

If $\gcd(a,b) = 1,$ then why is the set of invertible elements of $\mathbb Z_{ab}$ isomorphic to that of $\mathbb Z_a\times \mathbb Z_b$? I know the proof that as rings, $\mathbb Z_{ab}$ is congruent ...
2
votes
1answer
37 views

Every ideal is contained in a prime ideal that is disjoint from a given multiplicative set

Let $R$ be a ring $I\subset R$ an ideal and $S\subset R$ be a set for which holds: $1)$ $1\in S$ 2) $a,b \in S\Rightarrow a\cdot b\in S$ Show that there exists a prime ideal $P$ in $R$ containing ...
5
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1answer
602 views

What is the connection between the definition of complete intersection variety and complete intersection ring?

An algebraic variety is called a complete intersection if its defining ideal is generated by codimension many polynomials. A Noetherian local ring $R$ is called a complete intersection if its ...
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0answers
11 views

find the number of sub fields

Let $\omega$ be a complex number such that $\omega^3 =1$ and $\omega \not=1$. Suppose L is the field $Q(cuberoot{2}, \omega)$ generated by cuberoot{2} and $\omega$ over the field Q of rational ...
5
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1answer
54 views

Pronunciation of `Rng` - the non-unital Ring

I chuckled the first time I heard that a Ring without a multiplicative identity (Ring without the i) is called a ...
4
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1answer
90 views

Which rings containing the complex field are, as vector spaces over that field, isomorphic to $\mathbb{C}^2$?

Which rings $R$ containing (as a subring) the complex field $\mathbb{C}$ are, as a vector space over that field, isomorphic to $\mathbb{C}^2$? In other words: what are the two-dimensional unital ...
0
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1answer
42 views

Prove $\text{rad}(I)/I$ is isomorphic to $\mathfrak{N}(R/I)$

I want to know if this is the correct way to do it. Define $\varphi:\text{rad}(I) \longrightarrow \mathfrak{N}(R/I)$ by $\varphi(r)= r^n+I$,then ker$\varphi = I$, so therefore by the 1st isomorphism ...
0
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1answer
43 views

Prove every prime ideal of a ring is a radical ideal.

this is my attempt: Since $R$ is commutative, we let $I$ to be a prime ideal of $R$, the for $a,b\in R$,then the product $ab$ we must have that $a\in I$ or $b \in I$, by definition of a prime ideal. ...
2
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2answers
67 views

Show that $\mathbb{Q}( \sqrt2) \neq \mathbb{Q}( \sqrt3)$

The way that I'm thinking is by showing that the field extension $\mathbb{Q}( \sqrt2) /( \sqrt3) \neq \mathbb{Q}( \sqrt3)$, but is there a simpler way I'm ignoring?
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2answers
70 views

What is $\mathbb{F}_7[X]$?

I do not understand what sets like these are. I know what something like $\mathbb{Z}_7$ is. It is the ring of integers modulo 7 so it is equal to ${0,1,2,3,4,5,6}$. But what is $\mathbb{F}_7[X]$ equal ...
0
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2answers
37 views

In ring theory, what does $R^{2} \neq \{0\}$ mean?

I'm working on an exercise of Malik's Fundamentals of Abstract Algebra, namely: "Let $R$ be a ring such that $R^{2} \neq \{0\}$. Prove that $R$ is a division ring if and only if $R$ has no nontrivial ...
0
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1answer
23 views

Proof that set of units of a ring is a multiplicative group

How can I proof that the set of units of a ring is a multiplicative group? If I look at $\mathbb{Z}$ I have $\mathbb{Z}^*=\{-1,1\}$ Is it sufficient to say that $\mathbb{Z}^*$ is a subset of ...
0
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0answers
6 views

semisimplity of endomorphism of finitley generated module

Let M be a finitely generated left R-module and E = End(RM). Show that if R is semisimple (resp. simple artinian), then so is E.
0
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1answer
29 views

Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
2
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1answer
36 views

Questions on the field extension $K = \mathbb{Q}[x]/\langle x^2 − 5\rangle$

Given the field extension $K = \mathbb{Q}[x]/\langle x^2 − 5\rangle$ of $\mathbb{Q}$, and letting $a = [x] ∈ K$; 1) Show $K ≃ \mathbb{Q}(\sqrt5) $ and $[K : \mathbb{Q}] = 2.$ 2)Find the ...
2
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3answers
56 views

Explain to me the difference between the notation $\mathbb{Q}( \sqrt2) $and$ \mathbb{Q}[ \sqrt2]$

Please explain to me the difference between the notation $\mathbb{Q}( \sqrt2) $and$ \mathbb{Q}[ \sqrt2]$. I know that these two fields are equal. But what difference do the different brackets imply? ...
0
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0answers
21 views

Proofs with algebraic structures (rings)

If one is given a ring $R$ with a unity $u$, what are the steps one would have to take to prove that some element of $R$ named $s$ has a multiplicative inverse, where $-s$ also has a multiplicative ...
2
votes
1answer
30 views

Trace of nilpotent matrix over a ring

Let $R$ be a commutative ring with unity, and $n$ a positive integer. Let $A\in \mathfrak{M}_n(R)$ such that there exists $m\in \mathbb N$, for which $A^m=0$. Is it true that there exists $\ell\in ...
0
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0answers
20 views

Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
1
vote
1answer
23 views

Ideal of direct sum of rings

If $R$ is a ring and $S_1, S_2$ are subring of $R$ such that $R=S_1\oplus S_2$, Is there any relation between ideal of $R$ and Ideal of $S_1,S_2$? In particular I mean under which condition we can ...
1
vote
1answer
38 views

Prove that any subfield of $\Bbb R$ contains $\Bbb Q$

Prove that any subfield of $\Bbb R$ must contain $\Bbb Q$. Now for any subfield $F$ of $\Bbb R$, $1\in F$ so, $\Bbb Z \subset F \Rightarrow \Bbb Q \subseteq F$. Have I done it correctly?
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0answers
21 views

Why most discussion on ring(like module) is over PID, not UFD? [on hold]

I think many properties and discussion are based on unique factorization. Like On UFD, irreducible element is also prime. So why these analysises are not based on UFD?