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
70 views

Induction troubles proving a formula.

I'm having a tough time proving the following formula. Suppose $a,b\in R$ a ring. Define $a^{(0)}=a$, $a^{(1)}=[a,b]\equiv ab-ba$, and then $a^{(k)}=[a^{(k-1)},b]$. Then $$ \sum_{i=0}^k ...
1
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1answer
187 views

Principal ideal ring analytic functions

Could someone sketch a proof and explain me in words, why the set of analytic functions on $\mathbb{C}$ does not form form a principal ideal ring? Thank you!
1
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1answer
154 views

Every nonzero $x\in\mathbb{Z}[\sqrt{35}]$ belongs to finitely many ideals

How can one show that every nonzero element $x$ of the ring $\mathbb{Z}[\sqrt{35}]$ is contained in finitely many ideals? It is obvious in case of $x$ being invertible, but a general case is out of my ...
2
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1answer
207 views

Ring of analytic functions on the circle

Let $A = C^\omega(S^1)$ (resp. $C^\omega_{\mathbb C}(S^1)$) the ring of real-analytic real-valued (resp. complex valued) functions on the circle. These rings have maximal ideals $\mathfrak m_p = ...
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2answers
766 views

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
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2answers
219 views

Radicals of homogeneous ideals over semigroup-graded rings.

In this post I set out to prove an equation holding in a ring $R$ graded over $\mathbb{Z}$ or $\mathbb{N}$. The equation was: $\sqrt{J^\ast}=(\sqrt{J})^\ast$, where $J$ is an ideal of $R$, $J^\ast$ ...
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1answer
66 views

Algorithm to find nearest quotient in $\mathbb{Z}[i]$

Given two Gaussian integers $x$, $y$ what's the fastest way to find the Gaussian integer $z$ which minimizes $|x - zy|$? Then this Gaussian integer can be taken as $z = x/y$.
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2answers
530 views

Krull dimension of local Noetherian ring

Let $R$ be a commutative local Noetherian ring and $\mathfrak{m}$ its maximal ideal. Prove that, if $\mathfrak{m}$ is principal, then $\mathrm{dim}(R)\leq 1$ (the Krull dimension of the ring). ...
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2answers
744 views

Jacobson radical of $R[X]$, where $R$ is domain

Let $R$ be a commutative domain. Prove that the Jacobson radical of $R[X]$, i.e. the intersection of all maximal ideals, is the zero ideal. Thank you.
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1answer
220 views

Reduced ring is integrally closed in polynomial ring

Let $R$ be a commutative ring, with 1. Prove that if $R$ is reduced, then $R$ is integrally closed in $R[X]$, i.e. $R \subset R[X]$ is an integral extension of rings. I found this problem in many ...
4
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1answer
285 views

Is every proper nontrivial ideal in a Noetherian ring not flat?

I guess my general question is exactly what's in the title, but let me explain why I'm asking and how I came to it. Consider the ideal $I=\langle x,y \rangle \subset k[x,y]$ for a field $k$. Just to ...
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3answers
370 views

Localization in a ring

I am having trouble with the following problem. Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(-1 + ax)$. I ...
4
votes
1answer
107 views

What algorithms are there for determining whether a Gaussian integer is prime?

Give a Gaussian integer $z\in{Z[i]}$, how can I determine if $z$ is prime? I imagine there exists an algorithm that maps primality in $Z[i]$ to primality in Z. And for the case when $z\in{Z}$ I think ...
4
votes
4answers
858 views

product of two ideals

The product of two ideals is defined as the set of all finite sums $\sum f_i g_i$, with $f_i$ an element of $I$, and $g_i$ an element of $J$. I'm trying to think of an example in which $IJ$ does not ...
0
votes
2answers
283 views

Smallest ideal containing $I$ & $J$

I'm trying to think of an example of a ring $A$ and ideals $I$,$J$ s.t. $I \cup J$ is not an ideal. And what is the smallest ideal containing $I$ & $J$? Will $A \mathbb{Z}$, $I = 2\mathbb{Z}$, ...
4
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1answer
283 views

A formula for the minimum number of generators of a module over a semilocal ring

Let $R$ be a commutative ring with only finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_r$. Let $M$ be a finitely generated $R$-module. Then $$\mu_R(M)=\max\{\dim_{R/\mathfrak ...
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3answers
827 views

When the localization of a ring is a field

Let $R$ be a commutative noetherian ring with no nonzero nilpotents. Let $p$ be a minimal prime of $R$. Could you help me to prove that $R_p$ is a field?
2
votes
1answer
108 views

An equivalent condition for having finite length

Let $R$ be a commutative noetherian ring, $M$ a finitely generated $R$-module. How can I prove that $M$ has finite length if and only if $M_p=0$ for every non-maximal prime ideals $p$?
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2answers
1k views

Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain

We're proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain, using the norm function $$\nu (a + b\sqrt{2} ) = |a^2 - 2b^2|$$ and the first part says that since $\nu (a + b\sqrt{2} ) = |(a + ...
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0answers
110 views

Norm of the generators of a fractional ideal.

Let $\mathcal{O}_l=\mathbb{Z}[\frac{1+\sqrt{-l}}{2}]$ with $l$ a prime number congruent to 3 mod 4. Let $\mathfrak{a}$ be a non-principal fractional ideal of $\mathcal{O}_l$. My questions are: Why ...
2
votes
3answers
71 views

If $Ra$ is free for $a\neq 0,$ is $a$ regular?

Let $R$ be a commutative ring with unity, and $0\neq a\in R.$ We will say that an element $x\in R$ is linearly independent if $\{x\}$ is a linearly independent set. A non-zero element of $R$ is called ...
2
votes
2answers
166 views

Why is any irreducible ideal prime in a Boolean ring?

I want to prove the following: Let $A$ be a Boolean ring and let $\mathfrak{a}\neq A$ be an irreducible ideal. Then $\mathfrak{a}$ is maximal. I already know that the prime ideals of $A$ are ...
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2answers
194 views

Factoring a ring homomorphism

From Atiyah-Macdonald, bottom of page 9: "Let $f: A \to B$ be a ring homomorphism. ... We can factorize $f$ as follows: $$ A \xrightarrow{p} f(A) \xrightarrow{j} B$$ where $p$ is surjective and ...
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4answers
616 views

Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$

Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$ Where $N$ is the norm function that maps $\gamma = a+b\sqrt{n} \mapsto \left | a^2-nb^2 \right |$ I ...
2
votes
1answer
188 views

Is every semi-simple ring a product of simple rings

I was wondering if the following statements were true; 1) Every semi-simple ring is a product of simple rings. 2) Every module over a division ring $R$ is free. I think both of these statements are ...
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3answers
226 views

Showing that the elements $6$ and $2+2\sqrt{5}$ in $\mathbb{Z}[\sqrt{5}]$ have no gcd

In showing that the elements $6$ and $2+2\sqrt{5}$ in $\mathbb{Z}[\sqrt{5}]$ have no gcd, I was thinking of trying the following method. If the ideal $(6)$ + $(2+2\sqrt{5})$ is not principle in ...
2
votes
4answers
243 views

Showing that the only units in $\mathbb{Z}[i]$ are $1,\, -1, \, i, \, -i$?

How would you show that the only units in $\mathbb{Z}[i] :=\{a + ib \, |\, a,b\in \mathbb{Z}\}$ are $1,\, -1, \, i, \, -i$?
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4answers
201 views

If two elements in a ED have the same Euclidean norm, they are associates?

Is it very obvious that on a Euclidean Domain, two elements $x$ and $y$ have the same Euclidean norm $\nu(x) = \nu(y)$ then they are associates? Can someone give me a proof of this?
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2answers
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Is every Noetherian module finitely generated?

I was just wondering whether the following statement is correct. Let R be a ring and M a noetherian R module. Then M is finitely generated.
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3answers
565 views

Is $\lhd$ common notation for “is an ideal of”?

This question is because of this comment. I would like to know if I should refrain from using $\lhd$ for "is an ideal of" in ring-theoretic questions. Is it common enough, or should I explain what it ...
6
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1answer
227 views

If $R$ is commutative, and $J\lhd I\lhd R,$ does it follow that $J\lhd R?$

$\lhd$ will stand for "is an ideal of" in this post. Let $R$ be a commutative ring, $J\lhd I\lhd R$. Does it follow that $J\lhd R?$ I don't think it does, but I'm having difficulty finding a ...
4
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1answer
242 views

invertible polynomials over non-commutative rings

Let $f = a_0 + a_1 t + \dotsc + a_n t^n$ be a polynomial over some nontrivial, possibly noncommutative ring $R$. When is $f$ invertible in $R[t]$? When $R$ is commutative, the answer is well-known: ...
6
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2answers
297 views

Why is $1+a(t-1)$ a unit?

Let $R$ be a (noncommutative) ring and $a \in R$ such that $a(1-a)$ is nilpotent. Why is $1+a(t-1)$ a unit in $R[t,t^{-1}]$? Probably one just has to write down an inverse element, but I could not ...
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3answers
579 views

Can non-units be multiplied together to form units?

An irreducible element of a ring is a non-unit such that it cannot be written as a product of two non-units. A UFD is a domain such that each non-unit $x\in R \backslash \{0\}$ can be written as a ...
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3answers
184 views

What does the concept of a 'unit' mean in the set of polynomials?

If $\alpha$ is an algebraic element of $\mathbb{C}$, then there is a unique non-zero polynomial $f \in \mathbb{Q}[x]$ with leading coefficient $1$ such that $f(\alpha) = 0$, and $f$ is irreducible. ...
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1answer
517 views

An ideal maximal among those avoiding a multiplicative set is prime

Let $S$ be a multiplicatively closed subset of a ring $R$, and let $I$ be an ideal of $R$ which is maximal among ideals disjoint from $S$. Show that $I$ is prime. If $R$ is an integral domain, ...
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2answers
224 views

A maximal ideal among those disjoint from a multiplicative set is maximum [closed]

In a ring $R$, if $S$ is a multiplicatively closed set excluding $0$... letting $X$ be the collection of ideals disjoint from $S$, if $I\in X$ maximal, $J\in X$, prove that $J\subset I$.
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0answers
170 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
1
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1answer
237 views

Finding all simple $R$ modules of a ring.

I was hoping someone had an idea on how to go about solving the following; Find (up to isomorphism) all simple R-modules where i) $R = \begin{pmatrix} \mathbb{Z}/15 \mathbb{Z} & \mathbb{Z}/15 ...
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2answers
2k views

Shorter proof of $R/I$ is a field if and only if $I$ is maximal

Here is a proof I saw somewhere of the fact $R/I$ is a field if and only if $I$ is maximal: $\implies$ Suppose that $R/I$ is a field and $B$ is an ideal of $R$ that properly contains $I$. Let $b \in ...
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5answers
962 views

If the localization of a ring $R$ at every prime ideal is an integral domain, must $R$ be an integral domain?

Let $R$ be a commutative ring. Suppose that for every prime ideal $p$ of $R$, the localized ring $R_p$ is an integral domain. Must $R$ be a integral domain? I was trying to think of counter-examples, ...
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4answers
547 views

An ideal that is maximal among infinitely-generated ideals is prime.

I've been doing some old exam problems and I've come across a problem that I've answered, but my gut is telling me that there's something I'm glossing over. Let $R$ be a commutative ring with ...
8
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3answers
1k views

If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
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2answers
4k views

Khan academy for abstract algebra

I am looking for instructional videos for abstract algebra, specifically topics including group theory, ring theory, isomorphic and homomorphic structures, and properties of groups and rings, and ...
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4answers
306 views

Alternative proof of $I$ maximal implies $I$ prime

The proof I know to show maximal implies prime for an ideal $I$ in a commutative ring $R$ goes as follows: $I$ maximal $\iff$ $R/I$ is a field $\implies$ $R/I$ is an integral domain $\iff$ $I$ is ...
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2answers
185 views

On the sum of all the simple submodules of a module

$R$ ring and $M$ a left $R$-module. Call $\mathrm{Soc}\;M$ the sum of all the simple submodules of $M$. Then $M$ is artinian if and only if $\lambda_R(\mathrm{Soc}(M))<\infty$ and for very $0\neq ...
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2answers
180 views

If $M$ is a flat module then those two conditions are equivalent

Let $M$ be a flat $R$-module. Then the following are equivalent: 1) for every $R$-module $N$ we have $M\otimes_R N\neq0$ 2) for every maximal ideal $m$ of $R$ we have $M\neq mM$ I did 1 implies 2. ...
2
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1answer
137 views

On the length of a ring

Suppose that $R$ is a ring, and suppose that $\lambda_R(_RR)<\infty$ and $\lambda_R(R_R)<\infty$ (where $\lambda_R$ is the length of an $R$-module). Is it true that then ...
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0answers
51 views

What are 'symmetric (or 'new age') ring structures'?

The title says it: what are 'symmetric (or 'new age') ring structures'? The phrase was found in: Frank Quinn, Contributions to a Science of Contemporary Mathematics (Draft October 4, 2011), p 74 ...
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1answer
288 views

Ring notation, $R=\mathbb{Z}[1/N]$

Regarding the notation $R=\mathbb{Z}[1/N]$, where $N$ is a positive integer, does $R$ refer to: $R=\{a+b/N|a,b\in\mathbb{Z}\}$ or $R=\{a_0 +a_1/N+a_2/N^2+\ldots +a_n/N^n|a_i\in\mathbb{Z}\}$ or ...