An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ...

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Is there a general method to find if ideal is maximal

Is there an algorithm to determine if we have been given a ring $A$ and its ideal $I$, whether or not $I$ is a maximal ideal of $A$? I found that sometimes proving that ideal is maximal might be ...
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30 views

Showing that $A^*$ is and ideal and that “$*$” is multiplicative.

Let $E/F$ be an extension of algebraic number fields and $\mathcal{O}_E$ and $\mathcal{O}_F$ be the ring of integers. Define $$A^*=\mathcal{O}_EA,$$ where A is an ideal of $\mathcal{O}_F$. Prove that ...
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37 views

Generalize a trick with Dirichlet series to algebraic number theory

I am not able to generalize the following equality involving Dirichlet series : ...
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Aside from $\langle 0 \rangle$, can a ring of algebraic integers have prime ideals that are not maximal?

I have a feeling that a ring with such ideals would have to be non-UFD, and I can prove that in $\mathbb{Z}$ there are no such ideals. But in other rings, I'm not so sure. I'm not yet at a point at ...
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Is every non-trivial ideal in a commutative ring is a principal ideal?

I'm a bit lost... it seems every non-trivial ideal in a commutative ring is a principal ideal. but is it true? if not, could you pls give a counter example?
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Given distinct maximal ideals $M_1,…,M_n$, is $M_1\cdots M_n$ radical?

Let $R$ be a commutative ring with $1$ and $M_1,...,M_n$ be distinct maximal ideals in $R$. What I want to show is $M_1\cap\cdots \cap M_n=M_1\cdots M_n$. If I can show that $M_1\cdots M_n$ is ...
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Why the ideals here are in this form?

In this article, in the proof of problem no. 6, p. 3, it listed all the possible ideals, because they contains at least one of $2$, $3$, $5$, from $120$. And at least one of $x+1$, $x^2-x+1$. But I ...
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40 views

How to learn ideals and quotient rings?

I have difficulties to learn ideals of ring and how to operate with them. Is there somewhere a good tutorial on those? Like I saw from an algebra book the Artin–Rees lemma and it looked a bit scary as ...
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36 views

Intersection of localizations of an integral domain

I have a few questions about proving the following identities: $$\bigcap_{p \in SpecA}A_p = A \ \ \ \ \bigcup_{p \in SpecA}A_p = K$$ Here $A$ is an integral domain, $K$ is its field of fractions. ...
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69 views

Associated non-minimal prime ideal

I am trying to find an example of a noetherian local ring with an associated prime of height greater or equal 1. That is, I want a noetherian local ring $R$ together with an associated prime $p$ ...
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62 views

Ideals for commutative ring and equivalent statements

I need help solving a problem I have. Let $R$ be a commutative ring. Prove that for the ideals $I$ and $J$ of $R$ the following two conditions are equivalent. (a) The function $R\to R/I\times ...
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Ambiguity in the definition of unmixed ideal

Compare the definitions: Page 136 Matsumura, Commutative ring theory: A proper ideal $I$ in a Noetherian ring $A$ is said to be unmixed if the heights of its prime divisors are all equal. ...
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Is an ideal prime when its complex extension is prime?

Let $I = \langle f_1,\dots,f_k\rangle$ be an ideal in $\mathbb R[x_1,\dots,x_n]$. The same $f_i$ generate an ideal $\widetilde I$ in $\mathbb C[x_1,\dots,x_n]$. When $\widetilde I$ is prime in ...
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74 views

Radical Ideal for algebra

Show the following: a) $\text{rad}(IJ)=\text{rad}(I\cap J)=\text{rad}(I) \cap \text{rad}(J)$ b) $\text{rad}(I)=R$ if and only if $I=R$ c) if $P$ is prime $\text{rad}(P^n)= P$ for all $n$ d) Let ...
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36 views

Prove that $U = t · \mathbb{R}[t]$ is a maximal ideal in $\mathbb{R}[t]$

I was studying for an exam and chanced upon this question in my textbook. I was a bit confused as to how we would go about trying to solve it. Any help would be appreciated! :) Prove that $U = t · ...
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79 views

Show that $Rad(I)$ is a prime ideal

The ring $R$ is commutative with unit. An ideal $I$ is called primary, if it stands the following: If $ab \in I$ then $a \in I$ or $b^n \in I$, for a natural number $n$. Show that if $I$ is a ...
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138 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 ...
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$x$ 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 ...
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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 ...
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When is the quotient ring of a multivariable polynomial ring over a field by a monomial ideal an integral domain?

When is the quotient ring of a multivariable polynomial ring over a field by a monomial ideal an integral domain? I am actually trying to show that a monomial ideal is prime by showing the ...
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136 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 prime ideal $Q$ such that $0 \subsetneq Q \subsetneq P$. ...
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50 views

Number Theory: Class Groups

I have the following question regarding class groups. Show that $K=\mathbb{Q}(\sqrt{-19})$ is of class number 1. From what I understand, the Minkowski bound says, for a number field $K$, ...
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Example of non-commutative ring with exactly 2014 two sided-proper ideals.

Find a non-commutative ring with exactly 2014 two sided-proper ideals. Find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have ...
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81 views

If a certain ideal is radical or not

Let $n \in \mathbb{N}$ and let $I_{n}$ be an ideal in the polynomial ring $\mathbb{C}[x_{1},...,x_{n}]$ with the following properties: $I_n$ is generated by a (finite) number of polynomials which ...
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25 views

Minimal right ideals

Let $I$ be a minimal right ideal of a ring $R$ with $1$. If $r\in R$, could we say that $rI$ is zero or a minimal right ideal? I assumed a right ideal $J$ in $rI$ and intersecting it with $I$ got a ...
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22 views

Fractional ideals of $\mathbb{Q}$ prime to $N$

Let $N \in \mathbb{Z}$. What is meant by a fractional ideal $\mathfrak{p}$ of $\mathbb{Q}$ being prime to $N$? Is it that $gcd(\mathfrak{p},N\mathbb{Z})$ contains $\mathbb{Z}$? Let $I_N$ denote the ...
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170 views

algebra with topology homework problem

Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have: $$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists ...
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65 views

Carlson's translatability - are theses characterisations equivalent?

Given a translation-invariant ideal $\mathcal{I}$ on a commutative group $G$ and it's dual filter $\mathcal{I}^*$, I am trying to show that $$ (\forall I \in \mathcal{I})(\exists I' \in ...
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27 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 ...
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34 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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35 views

How to show that an ideal of $F[x]$ containing an irreducible polynomial of degree $n$ and a nonzero polynomial of degree $<n$ is $F[x]$?

Let $F$ be a field and suppose that $I$ is an ideal of $F[x]$ which contains an irreducible polynomial of degree $n$ and a nonzero polynomial of degree less than $n$. Show that $I=F[x]$. I can't ...
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Showing that an ideal is principal.

I need to show that the ideal $(3 + i , 6)$ is principal in $\mathbb{Z}[i]$ and find its generator. So I know that I need to find an element such that $<t> = (3 + i , 6)$. My intuition tells me ...
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39 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 ...
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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 ...
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Comparing an ideal and its saturation

Let $S = k[x_0,x_1,\ldots,x_n]$ with its usual grading and let $I \subset S$ be a homogeneous ideal not containing $S_+ = (x_0,x_1,\ldots,x_n)$. We define the saturation of $I$ to be the homogeneous ...
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$I$ 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?
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Factoring isomorphism

I have $\mathbb Z[i\sqrt2] = ${$a+bi\sqrt2; a,b \in \mathbb Z, i^2=-1$} and $I =\{a+bi: a,b \in \mathbb Z, i^2=-1, 11\mid a+3b\}$. My task was to prove that $I$ is an ideal in $\mathbb Z[i\sqrt2]$ by ...
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Quotient ring of the ring of integers of an algebraic number field and its fraction field

Let $K$ be an arbitrary algebraic number field. We know that the fraction field of $\mathcal{O}_K$ is $K$ which is always isomorphic to some $\mathbb{Z}[x]/(f(x))$. $\mathcal{O}_K$ also has dimension ...
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1answer
63 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 ...
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About images of (prime) ideals under injective endomorphisms

Let $f : R \to R$ be an injective unitary endomorphism of a commutative ring with 1. Let $I$ be an ideal of $R$. I have several related questions concerning the image of $I$ under $f$: 1) Under which ...
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How to show that if $\gamma \alpha=0$ (mod $A$) then $\alpha = 0$ (mod $A$)

Let $A$ be an ideal of $\mathcal O$ (Ring of integers of some algeibraic number field) and assume that $gcd([\gamma],A)= [1]$. How to show that if $\gamma \alpha=0$ (mod $A$) then $\alpha = 0$ (mod ...
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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 ...
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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. ...
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197 views

Prime ideals of the ring of integers of an algebraic number field

I am working on a problem that has a completely different point and I didn't work with algebraic number fields much before, so I was wondering if someone could point me in the right direction for ...
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56 views

Help unmasking a disguised principal ideal

I recently saw a question on here about trying to generate a non-principal ideal in a principal ideal domain, with the only answer so far saying that if the ring $R$ is a PID, then $\langle e, f ...
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Show that $V(I \cap J)=V(I) \cup V(J)$.

Let $I$, $J$ ideals of $K[x_1, x_2, \dots , x_n]$. I want to show that $$V(I \cap J)=V(I) \cup V(J)$$ I tried the following: $$\subseteq: $$ Let $x \in V(I \cap J)$. From the definition of $V$: ...
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72 views

Set of roots of sum is equal to the intersection

Let $(I_a)_{a \in A}$ be a family of ideals of $K[x_1,x_2, \dots, x_n]$. I want to prove that: $$V \left ( \sum_{a \in A} I_a\right )=\bigcap_{a \in A} V(I_a)$$ Do we have to use the definition: ...
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99 views

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$. I'm trying this problem with the Minkowski bound, please I need more help. Thanks
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81 views

Finding all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$

I want to find all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$. I know that $$\mathbb{Q}[x]/I \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x^2+x+1)$$ I ...
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62 views

In a PID, does every attempt to generate a non-principal ideal just lead back to the whole ring itself?

It is a well-known fact that a unique factorization domain is a principal ideal domain, in which all ideals are principal ideals. [EDIT: I got dyslexic on this one, should've said something along the ...