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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Prove that R is a field ↔ The only ideals in R are R(0) and (0). [duplicate]

Possible Duplicate: Rings and ideals The question is the title, any help would be greatly appreciated!
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A ring is a field iff the only ideals are $(0)$ and $(1)$

Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$. I can't figure out where to start other that I ...
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2answers
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Property of exact sequences?

If I have a commutative ring $R$ and an exact sequence $0\to M'\to M\to M''\to 0$ where $\epsilon:M'\to M$ and $\sigma:M\to M''$ do I get an exact sequence $0\to M'\to ...
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5answers
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Can an ideal of a ring be an ideal of a subring/superring too?

Let $R\neq S$ be rings with unity. Let $R$ be a subring (sharing the same unity) of $S.$ Let $\{0\}\neq K\subseteq R.$ Is it possible that $K$ is at the same time an ideal in $R$ and an ideal in $S?$
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1answer
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Needing a nudge with a Commutative Algebra Question

I have a commutative ring with identity $R$, and an $R$-module $M$. Next I have an $R$-submodule $N$ of $M$. Finally, I have a multiplicatively closed subset $S$ of $R$. I am asked to show that ...
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Localization of Quotient Groups

Suppose I have a commutative ring with identity $R$, and an $R$-module $M$. Next I have an $R$-submodule $N$ of $M$. Finally, I have a multiplicatively closed subset $S$ of $R$. An element $s\in S$ ...
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What are the integers $n$ such that $\mathbb{Z}[\sqrt{n}]$ is integrally closed?

I was recently reading about integral ring extensions. One of the first examples given is that $\mathbb{Z}$ is integrally closed in its quotient field $\mathbb{Q}$. Another is that ...
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3answers
570 views

How to determine whether a unique factorization domain is a principal ideal domain?

Could someone please provide an example of a unique factorization domain that is not a principal ideal domain? Furthermore, is there some way to determine whether a UFD is a PID?
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1answer
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R semisimple ring implies R direct sum of finite number of minimal left ideals.

I'm working through a proof of this statement which goes roughly as follows: "The simple submodules of $R$ (as a module over $R$) are exactly the minimal left ideals of $R$. So (from earlier ...
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1answer
222 views

Maximal subring

I have a quick question on a problem out of Dummit and Foote (problem 7.5.6 for those who might have the text): Prove that the real numbers, $\mathbb{R}$, contain a subring $A$ with $1 \in A$ and $A$ ...
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3answers
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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?
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1answer
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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 ...
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1answer
87 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 ...
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1answer
262 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 ...
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4answers
186 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. ...
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3answers
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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, ...
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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 ...
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3answers
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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 ...
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1answer
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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 ...
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1answer
158 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 ...
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2answers
370 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 ...
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1answer
354 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 ...
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1answer
138 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 ...
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1answer
254 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 ...
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1answer
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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 ...
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1answer
371 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 ...
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5answers
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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 ...
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3answers
893 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?
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1answer
92 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 ...
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2answers
627 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 ...
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2answers
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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 ...
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2answers
258 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] ...
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3answers
756 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 ...
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3answers
814 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. ...
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4answers
672 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 ...
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1answer
192 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 ...
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4answers
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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 ...
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2answers
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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 ...
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3answers
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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. ...
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1answer
258 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 ...
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1answer
131 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 ...
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1answer
360 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 ...
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0answers
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$A_1$ is a subring of $\operatorname{End}_{\mathbb{C}}(\mathbb{C}[x])$? [duplicate]

Possible Duplicate: $End_{\mathbb{C}} ( \mathbb{C}[x])$ and Weyl algebra Definition of $A_1=\left\{ \sum \limits_{i=0}^n f_i(x) \partial^i \mid n \in \mathbb{N}, f_i(x) \in ...
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2answers
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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 ...
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1answer
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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) ...
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2answers
193 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 ...
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1answer
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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 $R$ ...
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1answer
170 views

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

If $\zeta= \zeta_{n}$, how does one count the homomorphisms $f:\mathbf{Q}(\zeta)\rightarrow \mathbf{Q}(\zeta)$?
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1answer
71 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 ...
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1answer
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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 ...