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, ...

learn more… | top users | synonyms

0
votes
3answers
50 views

Prove that an ideal is not maximal

Ring $\mathbb Z[x],$ ideal is $(x)$. How to prove that this is NOT a maximal ideal? I can't imagine ideal, part of which would be $(x)$.
0
votes
2answers
36 views

Example of ideals such that $I^n=0$ but $I^{n-1}\not= 0$

Let $R$ be a ring. For each $n>0$ I want to find an ideal $I$ of $R$ such that $I^n=0$ but $I^{n-1}\not= 0$. Clearly this won't work for $R=\Bbb{Z}$ or $\Bbb{Z}/n\Bbb{Z}$. And I ran out of ...
1
vote
1answer
25 views

Showing that an epimorphism of an ideal is again an ideal

Let $R, S$ be commutative rings, $f : R \rightarrow S$ an epimorphism, I an ideal of R. Show that $f(I)$ is an ideal of $S$. As far as I understand, I need to show 4 things: 1) $0_s \in f(I)$ ...
2
votes
2answers
37 views

Alternative proof of '$I$ is maximal iff $R/I$ is a field'

For any commutative ring $R$ and an ideal $I$ of $R$, $I \neq R$, show that $I$ is a maximal ideal iff $R/I$ is a field. I write my own proof and it checks with the 'traditional' proof which ...
1
vote
1answer
33 views

Finding the ideal

Determine all the ideals, prime ideals, and maximal ideals of $\mathbb{R}[x]/I$ where $I$ is the ideal generated by $(x^2+1)(x-2)^2$. I am currently doing some reading on ideals (see ...
5
votes
1answer
37 views

Determine whether an ideal is principal or not

Let $I=\{a+b\sqrt{-3}: a+b \text{ even}\}$ be an ideal in $R=\mathbb{Z}[\sqrt{-3}]$. I want to determine whether $I$ is a principal ideal or not. I've been trying to work with the ideal $(2)$. I ...
0
votes
1answer
37 views

maximal ideal problem [duplicate]

I want to solve this problem, but I have no idea how I can start: If $K$ is a field, $(a_1,...,a_n) \in K^n,$ and $I$ the ideal $I=\langle x_1-a_1,...,x_n-a_n\rangle$, then how can we prove that ...
4
votes
2answers
42 views

Why $P_1\neq P_1P_2$?

Question: If $P_1,P_2$ are distinct prime ideals of an artinian ring, why is it that $P_1\neq P_1P_2$? I know that prime ideals of an artinian ring are maximal, but still, I can't see why ...
3
votes
2answers
74 views

The interpretation of ideals of a ring.

Ideals of a commutative ring (I have only studied the commutative case) are thought of as generalized numbers (in algebraic number theory) and as ring homomorphisms (through the ideal as kernel ...
0
votes
2answers
47 views

Confused on notions of maximal ideal and some notation

I'm just getting started learning ring theory and am currently learning about ideals. By book (Dummit & Foote) says the following: For example, in the ring $R = \mathbb{Z}[x]$ the elements $2$ ...
1
vote
1answer
29 views

one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...
4
votes
2answers
71 views

Ring whose all ideals are double-sided is commutative?

I was thinking about the following problem: Suppose R is a ring s.t. every left ideal is also right. Is R commutative? This actually continues the easier question: Suppose G is a group whose ...
0
votes
1answer
42 views

Intersection of distinct maximal ideals in a commutative ring with identity.

If $R$ is a commutative ring with identity and $M_1, \dots, M_r$ are distinct maximal ideals in $R$, then show that $M_1\cap M_2 \cap \cdots \cap M_r = M_1M_2\cdots M_r$. Is this true if "maximal" is ...
2
votes
2answers
35 views

$R$ is a ring. Prove that $R/(0_R)\cong R$

$R$ is a ring. Prove that $R/(0_R)\cong R$. I don't quite understand what $R/(0_R)$ looks like. By definition of quotient ring, it should have cosets $a+(0_R)$ where $a\in R$. So $R/(0_R)$ and ...
1
vote
1answer
14 views

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $.

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $. Help me some hints. Thank you in advance.
2
votes
1answer
32 views

prove that this ideal is radical

Let $A=\mathbb k[x,y,z]$ and let the ideal $$ I=(z-1,x^2-y).$$ I need to find $rad(I)$ but i don't know how. I think that this ideal is radical but I don't know good criteria for doing that =(
2
votes
1answer
34 views

Ideals agreeing in a localization

I have an integral scheme $X$, and two coherent ideal sheaves $\mathcal I$ and $\mathcal J$ on $X$. I know there is a (maybe not closed) point $x$ of $X$ such that $\mathcal I$ and $\mathcal J$ ...
0
votes
1answer
29 views

Infinite Varieties and Non constant Common Factors

I'm trying to work out some problems from Ideals, Varieties, and Algorithms, and I've stumbled on one that I'm unsure of how to start: Let $f,g \in \mathbb{C}[x,y]$ be nonzero. In this exercise, ...
2
votes
1answer
27 views

Maximal and prime ideals of $2 \mathbb Z$

What are the all maximal ideals of $2 \mathbb Z$ ? what are the all prime ideals of $2 \mathbb Z$ ? We know that if $R$ is a commutative ring with multiplicative identity $1$ and $M$ is a maximal ...
2
votes
1answer
45 views

Looking for example of a commutative non-unital ring in which every maximal ideal is a prime ideal

Give example of a commutative non-unital ring in which every maximal ideal is a prime ideal. The motivation for this question is : It is known that if $R$ is a commutative ring with identity $1 \ne ...
1
vote
3answers
34 views

Let $f$ be a surjective homomorphism. Prove that $\ker(f)$ is a maximal ideal

Let $f:R\to S$ be a surjective homomorphism, where $R$ is a commutative ring and $S$ is a field. Prove that $\ker(f)$ is a maximal ideal. I already know that $\ker(f)$ is an ideal of $R$. I tried to ...
1
vote
1answer
49 views

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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
25 views

Generalize a trick with Dirichlet series to algebraic number theory

I am not able to generalize the following equality involving Dirichlet series : ...
2
votes
1answer
26 views

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 ...
3
votes
1answer
37 views

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?
0
votes
2answers
22 views

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 ...
0
votes
1answer
24 views

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 ...
3
votes
1answer
32 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 ...
0
votes
1answer
25 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. ...
2
votes
1answer
74 views

Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$ and $\sqrt{IJ}\neq\sqrt{I}\sqrt{J}$ [closed]

Let $\Bbb k$ be a field and $I$, $J$ be ideals in $\Bbb k [x_1,x_2,\ldots,x_n]$. Show that: $(i)\; \sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$, $(ii)\; \sqrt{IJ}\neq\sqrt{I}\sqrt{J}$.
1
vote
0answers
60 views

Ideals, prime ideals and maximal ideals of the ring $K=\mathbb R[x]/\langle (x^2+1)(x-2)^2\rangle$ [closed]

I am trying to find the ideals, prime ideals and maximal ideals of this ring: $K=\mathbb R[x]/\langle (x^2+1)(x-2)^2\rangle$. I am fairly fluent in abstract algebra though ideals are my huge ...
4
votes
1answer
47 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 ...
0
votes
0answers
70 views

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. ...
2
votes
1answer
31 views

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 ...
0
votes
0answers
34 views

Radical Ideal for algebra

Show the following: a) rad(IJ)=rad(I∩J)=rad(I)∩rad(J) b) rad(I)=R if and only if I=R c) if P is prime rad(P^n)= P for all n d) Let F be a field and T a subset of F^n . Show that the ideal I(T) ...
0
votes
2answers
30 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 · ...
0
votes
1answer
46 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 ...
3
votes
2answers
57 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 ...
1
vote
2answers
16 views

$x$ 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 ...
2
votes
3answers
61 views

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 ...
0
votes
1answer
70 views

When is the quotient ring of a multivariable polynomial ring over a field an integral domain?

When is the quotient ring of a multivariable polynomial ring over a field an integral domain? I am actually trying to show that a monomial ideal is prime by showing the corresponding quotient ...
3
votes
1answer
66 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 $Q$ prime ideal such that $0 \subsetneq Q \subsetneq P$. ...
1
vote
1answer
39 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 $K$, ...
4
votes
1answer
64 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 ...
0
votes
1answer
20 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 ...
0
votes
0answers
40 views

Definition of $\sum_{a \in A} I_a$

If $(I_a)_{a \in A}$ a family of ideal of $K[x_1,x_2, \dots, x_n]$, I have the following definition in my notes: $$\sum_{a \in A} I_a=\{ a_{i1}+a_{i2}+ \dots+ a_{ij} | a_{ij} \in I_{a_j} \}$$ Is ...
0
votes
0answers
15 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 ...
5
votes
1answer
153 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 ...
0
votes
0answers
43 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 ...