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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8
votes
3answers
545 views

If $xy$ is a unit, are $x$ and $y$ units?

I know if $x$ and $y$ are units, in say a commutative ring, then $xy$ is a unit with $(xy)^{-1}=y^{-1}x^{-1}$. But if $xy$ is a unit, does it necessarily follow that $x$ and $y$ are units?
15
votes
1answer
422 views

Hopkins-Levitzki: an uncanny asymmetry?

Not every left Noetherian ring is left Artinian. Take $\mathbb{Z}$ as a quick example. But: Hopkins-Levitzki theorem: a left Artinian ring is left Noetherian. I find this quite amazing. I find ...
1
vote
1answer
90 views

Equivalent module categories

Let $A$ and $B$ be rings and let $A\text{-mod}$ and $B\text{-mod}$ be their abelian module categories. Let $F:A\text{-mod}\to M\text{-mod}$ and $F':B\text{-mod}\to A\text{-mod}$ be functors which ...
4
votes
1answer
287 views

Exercise on distributive module lattices

I'm trying to do the very first exercise in Representations and cohomology I by Dave Benson, it's been bugging me for a while now. I don't really know how to start, although I imagine we will need to ...
0
votes
4answers
189 views

Divisibility question for non UFD rings

Let $p$ be a prime element. I need an example of a domain in which $p^n$ divides $ab$ and $p^n$ does not divide $a$ and $p$ does not divide $b$. Obviously, the domain I'm looking for is not a UFD. ...
3
votes
3answers
148 views

Is $M_{2}(K)$ a polynomial identity ring?

Simply, if there is a polynomial $f$, in noncommuting variables, which vanishes under substitutions from ring $R$, the ring will be called a PI ring (Polynomial Identity ring). For example, ...
49
votes
5answers
3k views

Why “characteristic zero” and not “infinite characteristic”?

The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
0
votes
3answers
415 views

zero divisors and units for the group $\mathbb{Z}/n\mathbb{Z}$ with integer $n$

given the ring $ \mathbb{Z}/n\mathbb{Z} $ is always true that $ \mathbb{Z}/n\mathbb{Z}=[\text{zero divisor}]+[\text{units}] $ how can evaluate the zero divisor and units ?? I believe that $ a x=0 ...
3
votes
1answer
1k views

If $I$ is a maximal ideal of $R$, why is $R/I$ a field?

If $I$ is a maximal ideal of $R$, why is $R/I$ a field? I'm trying to use the fact that $I$ is maximal to show that $R/I$ only have ideals $\{0\}$ and $R/I$. Can anyone help me with this method. Many ...
6
votes
1answer
172 views

Integral closure in the total ring of fractions

My question is linked with normalization of reduced algebraic curves that are not necessarily irreducible. Let $(A,\mathfrak{m})$ be a local reduced noetherian ring with Krull dimension $1$, let ...
8
votes
2answers
438 views

Are ideals in rings and lattices related?

There are (at least) two notions of ideals: An ideal in a ring is a set closed under addition and multiplication by arbitrary element. An ideal in a lattice is a set closed under taking smaller ...
5
votes
1answer
357 views

Ring with subring isomorphic to $\mathbb{Z}$ and subring isomorphic to $\mathbb{Z}_{3}$

This is a homework question that I'm either not thinking through all the way, or I'm overcomplicating the issue. It reads Give an example of a ring that contains a subring isomorphic to ...
1
vote
1answer
140 views

Queries on proof that every PID is a factorisation domain

I'm reading a proof from C. Musili's Rings and Modules that every PID is a factorisation domain. The author defines a factorisation domain as a commutative integral domain $R$ with a unit such that ...
2
votes
1answer
272 views

Finitely presented modules

I know that one can compute Fitting ideals of a finitely presented module (over a commutative ring with identity). However, are they the only invariants of such a module? In other words, my question ...
1
vote
1answer
158 views

A question about semiabelian rings

Are these two definitions equivalent? A ring $R$ is called semiabelian by Yiqiang Zhou if its identity $1$ can be written as a finite sum $1 = e_1 + \cdots + e_n$ of mutually orthogonal idempotents ...
10
votes
1answer
391 views

Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
5
votes
5answers
1k views

Integral domain that is not a factorization domain

I am looking for rings that are integral domains but not factorization domains, that is rings in which it is not possible to express a nonzero nonunit element as a product of irreducible elements. Do ...
1
vote
3answers
1k views

When a prime ideal is a maximal ideal

In a commutative ring with unit every maximal ideal is prime. Under what conditions does the converse occur?
2
votes
1answer
94 views

Left ideal generated by $\lbrace ab-ba:a,b \in R \rbrace$ is a two-sided ideal

Let $R$ be a ring with $1$, and let $J$ be the left ideal of $R$ generated by $\lbrace ab-ba:a,b \in R \rbrace$. Then I want to show that $J$ is a two-sided ideal. I thought that since $J$ is a left ...
6
votes
2answers
762 views

$\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain

How can I prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain? Also, I need to prove that its field of fractions is isomorphic to the field of rational functions ...
2
votes
2answers
198 views

Can non-unital rings be replaced by R-algebras?

While working through some lecture notes on semigroups, it seemed to me like a semigroup doesn't buy you much generality over a monoid. But I wondered whether the situation is different for non-unital ...
2
votes
2answers
269 views

A few question in abstract algebra

I prepare my qualifying exams for my Ms.C. and I do a lot of exams, but a few problems in there, I couldn't resolve, I hope can you help me. 1) Prove the following ring isomorphism $$\mathbb{C}[x,y] ...
3
votes
3answers
858 views

Ring theory exercises at the graduate level

Do you know any book or an online source that contains exercises on ring theory? I've solved some exercises of Lang's Algebra and Dummit & Foote's Abstract Algebra but there is a huge gap between ...
6
votes
3answers
967 views

Dedekind domain with a finite number of prime ideals is principal

I am reading a proof of this result that uses the Chinese Remainder Theorem on (the finite number of) prime ideals $P_i$. In order to apply CRT we should assume that the prime ideals are coprime, i.e. ...
7
votes
4answers
769 views

Commutative rings without assuming identity

I was going through Exercises in Dummit&Foote, which does not assume identity in the definition of a ring, and reached the following exercise: Prove that in a Boolean ring ($a^2 = a$ for all ...
2
votes
1answer
196 views

Group Ring calculation

Let, $C_3=\langle\sigma\mid\sigma^3=1\rangle$, $a=\frac{1}{3}( 1+ \sigma +\sigma^2)$, $b=\frac{1}{3}(1+w \sigma+ w^2 \sigma^2)$ and $c=\frac{1}{3}(1+w^2\sigma+ w\sigma^2)$ where $w$ is the primitive ...
0
votes
4answers
106 views

there exist an extension such that this element is a zerodivisor?

Everyone knows that if in a ring A a unit a $\in$ A can´t be a zerodivisor. But could also be possible that "a" not be a zero divisor ( i.e does not exist a nonzero x $\in$ A , such that $ax=0$) but ...
9
votes
2answers
1k views

Methods to check if an ideal of a polynomial ring is prime or at least radical

I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example. To be more explicit, I am working ...
-1
votes
3answers
243 views

Can all rings with 1 be represented as a $n \times n$ matrix? where $n>1$.

It's just all the rings (with 1) I know can be written as a matrix, i.e., find some matrix representation of it (not necessary commutative). Complex numbers for example is written as obvious matrix. ...
2
votes
1answer
266 views

How can I prove that this polynomial is irreducible in $\mathbb{Q}[x]$?

How can I prove that $x^5+6x^3+x^2+3x+2$ is irreducible in $\mathbb{Q}[x]$? I tried with Eisenstein (also making the substitution $x\mapsto x-1$ and $x\mapsto x+1$ to see if I obtain an Eisenstein ...
1
vote
1answer
140 views

Dedekind domains

Let $A$ and $B$ be ideals. I want to show that there exists an element $c \in K$ (where $K$ is the quotient field of a Dedekind domain $O$) such that $cA$ is an ideal relatively prime to $B$. As ...
0
votes
1answer
466 views

First isomorphism theorem. How is this proof sufficient

I don't understand it. See for it to be isomorphism you need to it to be homomorphism between them. I can see how this is trivial from what's worked out. I can sort of agree that surjection ...
2
votes
2answers
114 views

How is $\mathbb{H}$ a $\mathbb{R}$-Algebra, but not a $\mathbb{C}$-algebra?

It says that $\mathbb{C}$ is not in the center of $\mathbb{H}$. Definition of $\mathbb{K}$-algebra for a ring if $Z(R)=K$. However, you can do this $(a+bI+cJ+dK)(e+fi)=(e+fi)(a+bI+cJ+dK)$. So I don't ...
1
vote
1answer
129 views

A question on the cokernel of an $R$-map between free $R$-modules where $R$ is an euclidean domain

Let $R$ be an euclidean domain, and $A$ a $m\times n$ matrix. I want to prove two things: 1) The torsion submodules of $\mathrm{Coker}\;A$ and $\mathrm{Coker}\;A^T$ are isomorphic. 2) ...
6
votes
2answers
195 views

Why do we assume $a+b=b+a$ in a Ring with 1. Also, is it true with Rings without 1?

I was wondering why do some people use redundant axioms in definitions? If you just expand $(a+1)(b+1)=(a+1)b+a+1=ab+b+a+1$ $(a+1)(b+1)=a(b+1)+b+1=ab+a+b+1$. Hence, $ab+a+b+1=ab+b+a+1$, then cancel ...
3
votes
1answer
89 views

Is a ring with the following properties semiprime? (Part 2)

Let $R$ be a ring with $1 \neq 0$ that contains noncentral idempotents. If for every noncentral idempotent $e$ of $R$ the corner ring $eRe$ is a division ring and $eR(1-e)Re \neq 0$, is the ring $R$ ...
3
votes
1answer
171 views

Counting endomorphisms of $\mathbf Q(\zeta _{n})$

If $\zeta= \zeta_{n}$, how does one count the homomorphisms $f:\mathbf{Q}(\zeta)\rightarrow \mathbf{Q}(\zeta)$?
2
votes
1answer
72 views

For $n\ge 3, x_{1},…,x_{n} \in \mathbf{Q}^{\ast}$, $[\mathbf{Q}(\sqrt{x_{1}},…\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$

For $n\ge 3, x_{1},...,x_{n} \in \mathbf{Q}^{\ast}$ and $[\mathbf{Q}(\sqrt{x_{1}},...\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$ how can we conclude that there are non empty $I \subset \{1,...,n\}$ with ...
4
votes
1answer
83 views

number of normed irreducible polynomials with degree d

I am Arthur from Belgium student in 2nd year of mathematics and I am repeating the exercises for Algebra I, but this one extra exercise I just can't solve: 14. ii) If $\mathbf{F}$ is a ...
2
votes
1answer
91 views

number of normed polynomials with degree d

I am Arthur from Belgium student in 2nd year of mathematics and I am repeating the exercises for Algebra I, but this one extra exercise I just can't solve: 14. ii) If $\mathbf{F}$ is a ...
5
votes
1answer
156 views

For which $n\in\mathbf{N}$ do we have $\mathbf{Q}(z_{5},z_{7}) = \mathbf{Q}(z_{n})$?

Put $z_{n} = e^{2\pi i /n}$. I am searching for $n \in \mathbf{N}$ so that $\mathbf{Q}(z_{5},z_{7}) = \mathbf{Q}(z_{n})$. I know that : $z_{5} = \cos(\frac{2\pi}{5})+i\sin(\frac{2\pi}{5}) $ and ...
4
votes
2answers
270 views

Application of Fermat's little theorem

I want to show this: $p$ prime, $p \equiv 2 \bmod 3 \implies t^{3}-a$ reducible in $\mathbf{F}_p[t]$ for all $a\in \mathbf{F}_{p}$. By using Fermat's little theorem, but I don't know how. If ...
4
votes
2answers
471 views

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$?

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$? If $\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\mid a,b,c,d\in\mathbf{Q}\}$ and $\mathbf{Q}(\sqrt{6})= \{a+b\sqrt{6} | ...
4
votes
1answer
228 views

What is the meaning of $\mathbf{Q}(\sqrt{2},\sqrt{3})$

I know that : $\mathbf{Q}(\sqrt{2}) = \mathbf{Q}+ \sqrt{2} \mathbf{Q}$ , but then what is $\mathbf{Q}(\sqrt{2},\sqrt{3})$?
5
votes
2answers
174 views

elegant way to show $P= t^{1024}+t+1$ is reducible in $\mathbf{F}_{2}[t]$

This is homework exercise: $$P=t^{1024} + t + 1 , R = \mathbf{F}_{2}[t] \Rightarrow P \ \text{reducible in R}$$ I wanted to show this analogous to how a book shows it (book shows it with other ...
4
votes
1answer
116 views

Finitely generated modules over a PID, and normed polynomials

I am Arthur from Belgium student in 2nd year of mathematics and I am repeating the exercises for Algebra I, but this one extra exercise I just can't solve: 14.$\text{}$ i) If $R$ is a PID, ...
6
votes
1answer
112 views

Showing a degree formula $\dim_{\mathbb{C}} R^{2} / L$

If $a,b,c,d$ are in $R=\mathbb{C}[t]$ and $ad-bc \ne 0$, $L= R(a,b)+R(c,d)$ in $R^{2}$. I want to show that $\dim_{\mathbb{C}}R^{2}/L = \deg(ad-bc)$. In a previous theorem it was shown that : ...
5
votes
1answer
461 views

Relation between primary ideal and prime ideal

We know that every prime ideal is primary ideal. But can we say, every primary ideal is a power of prime ideal? if it is not correct a counterexample. Thanks.
27
votes
3answers
2k views

Does every Abelian group admit a ring structure?

Given some Abelian group $(G, +)$, does there always exist a binary operation $*$ such that $(G, +, *)$ is a ring? That is, $*$ is associative and distributive: \begin{align*} &a * (b * c) = ...
2
votes
1answer
94 views

question about characteristic of idempotent.

If $e$ and $f$ are idempotents, in a ring $R$. My question is how I can show $Re+Rf=Re+R(f-fe)$. Thanks