Questions tagged [ideals]

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, ideals in Lie algebras.

8
votes
1answer
48 views

Constructing rings with a specific lattice of ideals.

Let $R$ be a commutative ring with 1. The ideals of $R$ form a lattice with inclusion as order relation. Let me call it the ideal lattice $L(R)$ of $R$. Given an arbitrary lattice $L$, there are some ...
0
votes
2answers
59 views

Short Question: $(p)$ for a prime is not a maximal ideal in $\mathbb{Z}[X]$

Given a prime number $p\in\mathbb{Z}$ I want to show that $(p)$ is not a maximal ideal in the ring of polynomials $\mathbb{Z}[X]$. I know how the maximal ideals in $\mathbb{Z}[X]$ look like, I want ...
2
votes
1answer
61 views

On the maximal ideals of $\Bbb Z_5[X,Y]$ which contain $\langle Y \rangle$

Let $R:=\Bbb Z_5[X,Y]$ and $I:=\langle Y \rangle \trianglelefteq R.$ 1) Prove that $I$ is prime but not maximal ideal. 2) Find all maximal ideals of $R$, which contain $I$. Answer. 1) If we take ...
0
votes
0answers
20 views

Counterexamples where $\sum_i (a_i : b) \subsetneq (\sum_i a_i :b)$ or where $\sum_i (a: b_i) \subsetneq (a : \bigcap_i b_i)$?

Better formatted title: Counterexamples where $\newcommand{\mf}[1]{\mathfrak{#1}}$$\sum_i (\mf{a}_i : \mf{b}) \subsetneq (\sum_i \mf{a}_i : \mf{b})$ and/or where $\sum_i (\mf{a}: \mf{b}_i) \subsetneq (...
3
votes
1answer
43 views

C(0, 1) has infinitely many prime ideals not of the form Mp [duplicate]

I know $\mathcal C([0, 1])$ has all maximal ideals of the form $M_p=\{f\in \mathcal C([0, 1]) : f(p) =0,\ p\in [0, 1] \}$. My question is little bit different. If I replace $[0, 1]$ by $(0, 1)$ ...
-1
votes
1answer
26 views

How do you prove that 2 nontrivial ideals are the only nontrivial ideals in a ring?

What do you need to look for in 9rder to prove that 2 nontrivial ideals are the only nontrivial ideals in the ring? I can prove that if my ideals are I and J that I+J=R, where R is the ring, but I don'...
0
votes
0answers
11 views

About countable boolean algebras

my question is concerning the article from book "countable boolean algebras and decidability", Goncharov. here we sat homomorphism from A to B image 1 here we define composition of ideals (I×J), and ...
0
votes
4answers
295 views

For an algebraically closed field $k$, an ideal $I$ of $k[x]$ is maximal if and only if $I = (x-c)$

This is an exercise $4.21$ on a page $155$ from a textbook "Algebra: Chapter $0$" by P.Aluffi. Let $k$ be an algebraically cloased field, and let $I \subseteq k[x]$ be an ideal. Prove that $I$ is ...
0
votes
1answer
35 views

Lemma for Chinese Remainder Theorem

I want to prove the following Lemma: Let $R$ be a commutative ring with $1\neq 0$ and $I,J_{1}, J_{2},..., J_{n}$ be the ideals of $R$ such that $I+J_{i}=R$ $\forall 1\leq i\leq n$. Then, $I$, $J_{1}...
1
vote
1answer
34 views

Understanding the Definition of Ideals Generated by Polynomials

I would like to confirm whether my interpretation of the definition of ideals generated by polynomials is correct, please. In Ideals, Varieties, and Algorithms, Cox et al. define this as such (...
0
votes
1answer
33 views

Inverse of a finitely generated ideal in UFD

Let $R$ be a UFD and $K$ be its field of fractions. Let $A$ be an ideal of $R$. Define for this $A$, the $R$-submodule $A^{-1}$ of $K$ given by $$A^{-1}=\{ \alpha\in K \,\,:\,\, \alpha A\subseteq R\}....
1
vote
0answers
30 views

Find an ideal $I$ in $A$ so that $A/I$ is a finite field of $25$ elements.

Let $A = \frac {\Bbb Z[X]} {\left ( X^4+X^2+1 \right )}.$ Find an ideal $I$ in $A$ such that $A/I$ is a finite field of $25$ elements. I have seen that the polynomial $X^4+X^2+1$ is reducible in $\...
0
votes
1answer
63 views

Converse of : If $M$ is Noetherian then $M_m$ is Noetherian for every $m\in\operatorname{Max}(R)$.

Let $R$ be a Noetherian ring and $M$ be an $R$-module. We know that if $M$ is Noetherian then $M_m$ is a Noetherian $R_m$-module for every $m\in\operatorname{Max}(R) $. Now, I want to know if the ...
5
votes
3answers
456 views

Set of associated primes of direct sum

Let $M$ be a module over a ring $R$. Let $\operatorname{Ass}(M)$ be the set of annihilator ideals $\operatorname{Ann}(x)$, which are prime, so $$\operatorname{Ass}(M) = \{\operatorname{Ann}(x) \mid \...
11
votes
1answer
2k views

Is each power of a prime ideal a primary ideal?

I want to show that each power of a prime ideal is a primary ideal or I have to think about a counterexample?
1
vote
1answer
19 views

For ideals of $R$, $I_1, I_2$, Is it true that $(I_1 \cap I_2)M \cong I_1 M \cap I_2 M$ for any $R$-module $M$?

It seems true if $M$ is finitely generated module over PID $R$ because we can take a linearly independent finite generator and so we can use $$ \sum_{i=1}^n c_i m_i = 0 \Leftrightarrow \forall i \in \{...
0
votes
1answer
31 views

Prove that $\left( I_1I_2 \right)^e=I_1^e I_2^e$

I have a problem when proving a basic exercise about extension in commutative ring: Let $f:A \rightarrow B$ be a ring homomorphism, $I_1$ and $I_2$ are ideals of $A$. Prove that $\left( I_1I_2 \...
1
vote
1answer
24 views

Decomposition of semi-simple Lie algebra into simple lie algebra (or ideal?)

A semi-simple Lie algebra $L$ can by definition be decomposed into simple Lie algebras : $L=L_1\oplus \ldots \oplus L_n $. Are these $L_i$ necessarily ideals of $L$?
1
vote
1answer
20 views

show that a kernel is principal, here the application : $ \text{ev}_{x = t^2, y = t^3} $

$\mathbb C [x,y] \to \mathbb C [t]$ with the evaluation : $$ \text{ev}_{x = t^2, y = t^3} $$ How can you show that the kernel is principal in $\mathbb C[x,y] $? I think the kernel is the polynomial ...
0
votes
2answers
28 views

$\mathfrak{q}$ is primary iff: given $a,b \in A$, $ab \in \mathfrak{q}$ and $a \not\in \mathfrak{q}$, then $b^n \in \mathfrak{q}$ for some $n \geq 1$

I am studying from Lang's Algebra, and in Chapter X Noetherian Rings and Modules, $\S$3 Primary Decomposition, he makes the following definitions on page 421, third edition (assume that $A$ is a ...
1
vote
1answer
51 views

Quadratic field ideal find $\mathbb{Z}$-basis given a $\mathcal{O}_K$-basis

Suppose we are working in an imaginary quadratic number field $\mathbb{Q}(\sqrt{d})$ (so $d$ is a fundamental discriminant with $d < 0$). Now in the ring of integers $\mathcal{O}_K$ suppose we ...
2
votes
0answers
52 views

why is a ring ideal not called a filter

A ring ideal can be characterized by the two rules: $$(a\in I) \wedge (a ~ \textrm{divides} ~ b) \implies b \in I$$ $$ a,b \in I \implies \textrm{gcd}(a,b) \in I$$ (the usual definition states $a,b \...
0
votes
0answers
9 views

For Pairwise Comaximal ideals $I_1, …, I_n$, $I_1 \cap…\cap I_n \subset I_1I_2 \cdots I_n$

In a set of Pairwise Comaximal $I_1, ..., I_n$ ideals of a commutative ring, $I_1 \cap...\cap I_n \subset I_1I_2 \cdots I_n$. I get how to do it for the case when $n=2$: choose $a \in I, b \in J,s.t. ...
0
votes
1answer
29 views

$(0)$ and $p^n\mathbb{Z}$ (where $p$ prime, $n$ positive integer) are the only primary ideals in $\mathbb{Z}$

I am trying to show that $(0)$ and $p^n \mathbb{Z}$ are the precisely primary ideals in $\mathbb{Z}.$ Clearly $(0)$ is a prime ideal hence primary and radical of $p^n \mathbb{Z}$ being the maximal ...
0
votes
2answers
40 views

In $\mathbb Z[x]$, is $(2,x)=(2)+(x)$?

The text says that $(2,x)=(2)+(x)$, because $1 \in \mathbb Z$. I do not see why this leads to the decomposition. Can someone point me in the right direction?
1
vote
0answers
21 views

Coloring of hypergraph with polynomial implication

Sorry to bother again with my misunderstandings, but I encountered yet again an issue with the topic of coloring in graphs and again I require some help to deal with this exercise: Let $\mathbb{F}$ ...
2
votes
3answers
45 views

Find an ideal $I$ of $\mathbb Z/2\mathbb Z[x]$, such that $\mathbb Z/2\mathbb Z[x]/I$ is a field of 8 elements.

My attempt: A field can be constructed by taking the quotient by a maximal ideal. $\mathbb Z/2\mathbb Z[x]$ is a PID, so a maximal ideal can be made by taking an ideal generated by an ...
6
votes
1answer
54 views

When do (multivariate) polynomial rings fail to be Prüfer rings?

In what follows, ring is defined to be a commutative ring with unit ($1$). Definition: Perhaps over-generalizing from (12) in this Math.SE answer, call any ring $R$ a Prüfer ring if, for all non-zero ...
0
votes
0answers
17 views

If all submodules of a free $R$-module is free, then $R$ is a PID. [duplicate]

Let $R$ be a commutative ring with identity. Prove that if all submodules of a free $R$-module is also free, then $R$ is a PID. I have managed to prove $R$ is an integral domain. For the PID part, by ...
0
votes
1answer
34 views

Finding monic generator for ideals generated by polynomials of $\mathbb{Q}[X]$

Let $R = \mathbb{Q}[X]$ be the ring of polynomials with rational coefficients and let: $$I = \langle X^2 + 1 \rangle = \{(X^2 + 1)f(X) \ \vert \ f \in\mathbb{Q}[X] \} $$ $$J = \langle 2X + 1,...
5
votes
1answer
238 views

Isotypic components are just simple two-sided ideals

I'm trying to show that when we decompose a semisimple ring $R$ into isotypic components $$ R \overset{_R\mathsf{Mod}}{\cong}\bigoplus_{j=1}^{k_1}{I^{(1)}_j} \bigoplus \dotsb \bigoplus \left( \...
2
votes
0answers
43 views

Ideal generated by two relatively prime polynomials

Let $f(x,y),g(x,y)\in\mathbb R[x,y]$ such that $\gcd(f(x,y),g(x,y))=1$. Let also $I\triangleleft\mathbb R[x,y]$ be the ideal generated by $f(x,y), g(x,y)$. $I=\langle f(x,y),g(x,y)\rangle$ What ...
0
votes
1answer
45 views

Nonzero prime ideal containing no other nonzero prime ideal

Let $R$ be a UFD and $P$ be a nonzero prime ideal of $R$. Suppose that $P$ does not contain any nonzero prime ideal other than $P$. What can you say about $P$? I think $P$ should be a principal ideal....
0
votes
1answer
119 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 \...
0
votes
1answer
34 views

Show that if $I$ is an ideal contained in $J$ then $(R/I)\setminus (R/I)^{\times}$ is an ideal of $R/I$

Question: Let $R\neq 0$ be a ring and $J= R\setminus R^{\times}$ an ideal of $R$. Show that if $I\subset J$ is an ideal of $R$ then $(R/I)\setminus (R/I)^{\times}$ is an ideal of $R/I$ Attempt: I ...
0
votes
0answers
6 views

The ideal generated by a maximal orthogonal system in a Banach lattice

I have a pretty specific question about H.H. Schaefer's "Banach lattices and positive operators" book. In chapter 3, part 6 (page 169), it is said that the ideal generated by a set S (which is a ...
0
votes
1answer
40 views

Find a finite Gröbner basis for ideal $I \subseteq \mathbb{R}[x, y, z]$

Find a finite Gröbner basis in lexicographic ordering $x \prec y \prec z$ for ideal $I \subseteq \mathbb{R}[x, y, z]$, where $$ I = \{ f \in \mathbb{R}[x, y, z] \space | \space f(a, -a, 2) = 0 \...
15
votes
5answers
3k views

Idempotents in a local ring

Is it true that a local ring, i.e., a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
1
vote
2answers
44 views

Why is the Killing form of $\mathfrak{g}$ restricted to a subalgebra $\mathfrak{a} \subset \mathfrak{g}$ not the Killing form of $\mathfrak{a}$?

I know that the Killing form of $\mathfrak{g}$ restricted to an ideal $I \subset \mathfrak{g}$ is just the Killing form of $I$. However, what happens in general if we relax the conditions and just ...
1
vote
0answers
28 views

Singularity of a curve in higher dimensions

Although the problem I am working on is regarding a curve in 6 dimensional affine space, for simplicity, let's consider the Viviani's curve in $\mathbb{C}^3$: $$\mathcal{I}=\langle {x}^{2}+{y}^{2}+{z}^...
0
votes
1answer
67 views

Ideals of $R[x]/I[x]$, where $I$ is a maximal ideal of $R$

Original Question: Let $R$ be a commutative ring with identity and $I$ maximal ideal in $R$. Show that $I[x]$ is a prime ideal in $R[x]$ and is not maximal ideal in $R[x]$, find two distinct maximal ...
0
votes
1answer
13 views

Is the image of an ideal under a surjective Lie algebra homomorphism an ideal?

Say we have $\phi: \mathfrak{g} \to \mathfrak{h}$, where $\phi$ is a surjective Lie algebra homomorphism. Is $\phi(I)$ an ideal of $\mathfrak{h}$? I think this holds because for every element of $h \...
3
votes
2answers
49 views

Prove $(y-x^2)$ is a prime ideal in $\mathbb{R}[x,y]$, but not maximal.

My guess is to use the fact that when we take the quotient, $\mathbb{R}[x,y]/(y-x^2)$, this will become an integral domain but not a field. I am not sure how to take the quotient, though. I am also ...
1
vote
1answer
44 views

Divisorial ideal ($v$-ideal) of $A=\mathbb{Z}+X\mathbb{Q}[[X]]$.

I know that the domain $R$ is Mori if and only if for every nonzero ideal $I$ of $R$, $I_v=(a_1,\dots,a_n)_v$ for some $a_1,\dots,a_n\in I$. My goal is to find an example satisfying : For any $v$-...
0
votes
1answer
22 views

How do we know a set can be ordered and how do we know how to form the order? Zorn Lemma

I was reading a lot of proofs that involves Zorn Lemma in Algebra. I give one example. All commutative ring $A$ with $1$ has at least one maximal ideal. The proof goes like this Set $\Sigma = \{...
3
votes
1answer
91 views

Is it still unknown whetever any $\mathscr{D}$-class $D$ being a semigroup is bisimple?

Introduction: It is stated in my book from year 1961 that it is unknown whatever a subsemigroup $D$ of semigroup $S$, which is also a $\mathscr{D}$-class of $S$ is always bisimple, but it is known ...
0
votes
2answers
34 views

Contraction of quotient ideal is quotient of contractions?

Let $\mathfrak a,\mathfrak b$ be ideals in a ring $A.$ The quotient of $\mathfrak a$ and $\mathfrak b$ is $(\mathfrak a:\mathfrak b)=\{x\in A:x\mathfrak b\subseteq \mathfrak a\}$ and if $f:B\to A$ is ...
0
votes
1answer
27 views

Finding a prime ideal whose class is order 10 in the ideal class group

"By factorising the ideal $(4+\sqrt{-74})_{R}$, or otherwise, find a prime ideal whose class $[P]$ in the ideal class group $Cl(R)$ has order 10" $R = \mathbb{Z}[\sqrt{-74}]$ So I've managed to get ...
4
votes
1answer
56 views

Ideal in polynomial ring extension [duplicate]

Let $K \subset L$ be a field extension, $K[X]$ and $L[X]$ the corresponding polynomial rings (in one variable) and $I \subset K[X]$ an ideal. I want to show that $I=K[X] \cap IL[X]$, where $IL[X]$ ...
1
vote
1answer
32 views

example of right / left ideals

I searched and searched for examples of right / left ideals, but could find none. I read that a right ideal of $S$ is a subset of $R$ of $S$ such that $RS \subseteq R$, and that symetrically, a left ...