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

1
vote
2answers
20 views

Let A and B be ideals of a ring and C a prime ideal. Prove if the intersection of A and B is a subset of C then either A or B is a subset of C

Claim: Let $A$ and $B$ be ideals of a commutative ring $R$ and $C$ a prime ideal of $R$. Suppose that the intersection of $A$ and $B$ is a subset of $C$. Prove either $A$ or $B$ is a subset of $C$. ...
1
vote
1answer
12 views

Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$, where $N_G = {\sum}_{g \in G} {g} \in F[G]$. [duplicate]

I am trying to solve this Representation Theory question: Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a ...
0
votes
2answers
34 views

Show that the ideal generated by $x^2-2$ is maximal

Let $A = \mathbb{Q}[x]$. Show that the ideal generated by $x^2-2$ is maximal. I think it is sufficient to show that $\mathbb{Q}[x]/(x^2-2) \cong \mathbb{Q}\sqrt{2}$, where $\mathbb{Q}\sqrt{2} = \{a + ...
0
votes
2answers
28 views

Is the ideal generated by the polynomial $x^2-2$ maximal in the ring $\mathbb{C}[x]$?

Personal question: Is the ideal generated by the polynomial $x^2-2$ maximal in the ring $\mathbb{C}[x]$? I know the ideal in $\mathbb{Q}[x]$ generated by $x^2-2$ is maximal, considering ...
2
votes
1answer
27 views

Why is $|V(I)| \leq d_1\cdots d_n$?

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a ...
1
vote
0answers
21 views

Minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$ [duplicate]

Write out a minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$, and determine the primes belonging to $I$. Determine the dimension of the ring $\mathbb C[X, Y, ...
-1
votes
2answers
40 views

Find an infinite collection of maximal ideals containing $(x^2 - y^3) \subset \mathbb{C}[x,y]$ [on hold]

What is an infinite collection of maximal ideals containing the ideal $I = (x^2 - y^3) \subset \mathbb{C}[x,y]$?
0
votes
0answers
12 views

Proof that maximal ideals in $\mathcal{P}[x_0]$ intersected with $\mathcal{P}$ is a maximal ideal in $\mathcal{P}$ [on hold]

I am trying to show that maximal ideals in $\mathcal{P}[x_0]$ intersected with $\mathcal{P}$ is a maximal ideal in $\mathcal{P}$, where $\mathcal{P}$ is the polynomial ring $K[x_1, \dots, x_n]$ or ...
0
votes
1answer
32 views

Quotient of maximal and prime ideals

Given that $I, J$ are ideals in $R$, $I$ is maximal or prime, do we have that $I/J$ is maximal in $R/J$? $I/J$ is prime in $R/J$? I think it is true but don't see how it works.
0
votes
1answer
68 views

Do there exist polynomials $f,g$ in $\mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$.

Do there exist two polynomials $f, g \in \mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$? I know that this cannot happen in $\mathbb{R}[x]$. However, since $\mathbb{C}$ is algebraically closed, this ...
0
votes
1answer
12 views

Principal Ideal Domain $R$ and ideal $J\neq 0$ so that that $R/J$ have a finite number of ideals. [duplicate]

Let $R$ a un Principal Ideal Domain(PID) and $J\neq 0$ a ideal of $R$. Show that $R/J$ have a finite number of ideals.
1
vote
0answers
28 views

When is the ideal generated by 2 elements equal to the sum of the 2 ideals

Is it true in general that (a,b)=(a)+(b)? I would suppose that (a)+(b)$\subset$(a,b) and i believe the reverse containment should hold as well, i just can't seem to fit the pieces together.
0
votes
0answers
12 views

Laurent Ideal whose Intersection with Polynomial Ring Requires More Generators

I want to find an ideal $I\subseteq \mathbb Q[x^{\pm 1}, y^{\pm 1}, z^{\pm 1}]$ which requires fewer generators than the affine ideal $I\cap \mathbb Q[x, y, z]$. I tried finding a principal ideal $I$ ...
0
votes
2answers
41 views

Prove that the sum of ideals of a ring A equals A and its intersection is zero.

I've been looking at a couple of ring theory exercises and there's this one I don't know how to do it. It goes like this. $A$ is a commutative unital ring, and $e$ an element of $A$, $e \neq ...
0
votes
1answer
18 views

Saturation of a multiplicatively closed subset

Exercise 3.7 of Atiyah-MacDonald asks the reader: if $A$ is a commutative ring and $\mathfrak{a} \triangleleft A$ an ideal, find the saturation of $1 + \mathfrak{a}$. Previously we have shown that ...
1
vote
1answer
28 views

Property of multiplication of ideals in $\mathcal{O}_K$

Let $\mathfrak{a}, \mathfrak{b}$ be two coprime ideals of $\mathcal{O}_K = \mathbb{Z}[\sqrt{-d}]$ such that $\mathfrak{a}\mathfrak{b} = (n)$ for some $n \in \mathbb{Z}$. Does $\mathfrak{a}^m = (u)$ ...
0
votes
1answer
44 views

Generators of the Tangent Space

Let $X$ be an affine variety, $X \subset A^n$ and suppose $f_1(T),\ldots,f_r(T) \in K[T_1,\ldots,T_n] $ generate $I(X)$. (Note that $I(X)$ is the ideal of $K[T_1,\ldots,T_n]$ of which elements of $X$ ...
3
votes
1answer
39 views

Is there any way to gain some insight into a proof by simply looking at a graphic?

My school is using Pinter's "A Book of Abstract Algebra" for both semesters of Modern Algebra. For a class assignment a couple weeks ago, regarding rings, I was tasked with the following problem ...
1
vote
1answer
21 views

If $X$ is maximal ideal then it consists of non-invertible elements?

I'm reading through a paper where I came across the following theorem Let $A$ be a commutative complex Banach algebra with unit element $e$. Theorem: A subspace $X\subset A$ of codimension $1$ is ...
2
votes
1answer
27 views

Essential ideals

I am trying to get my head around essential ideals. In literature I found 2 definitions: An ideal $I$ in a C*-algebra $A$ is essential in $A$ (i) if $aI = 0$ implies $a=0$, $a\in A$; or (ii) if ...
0
votes
0answers
21 views

Local banach algebra without zero divisors

I need to construct example of banach algebra with unique nontrivial maximal ideal and without zero divisors. I think it is must be a subalgebra of $\mathbb{C}[[z]]$, but I could not build anything.
1
vote
1answer
84 views

Computing the radical of $\mathfrak{gl}(2,\mathbb{C})$ without using the semisimplicity of $\mathfrak{sl}(2,\mathbb{C})$.

I have been trying to show that the radical of $\mathfrak{gl}(2,\mathbb{C})$ is its center, i.e. scalar matrices, however all the proofs I have encountered (e.g. Radical of $\mathfrak{gl}_n$) have ...
2
votes
1answer
47 views

Extended ideals and algebraic sets

Let $L\subset k$ a field extension such that $k$ is algebraically closed. Now consider the algebraic set $Z(\mathfrak a)$ where $\mathfrak a$ is an ideal of $k[T_1,\ldots, T_n]$ but it is generated ...
-1
votes
1answer
33 views

Maximal chain of primes in a finitely generated $\mathbb C$-algebra

Let $A=\mathbb{C}[x,y,z]/\langle xyz-1\rangle$. Find a maximal chain of primes in $A$. I think it has to do something with the Krull dimension but I don't really know how to construct such a ...
0
votes
1answer
46 views

Given $I,J$ ideals of $R$, show that $\forall m \geq 1; I^m + J^m =R$ [duplicate]

Given $R$ a commutative ring and $I,J$ ideals of $R$, such that $I+J=R$, show that $I^m+J^m=R, \forall m \geq 1$ My problem is that I don't know what is the meaning of $I^m$ and in literature I ...
1
vote
1answer
21 views

Ideal intersection in Boolean polynomial ring

I'm trying to calculate $ \langle x,y,z\rangle \cap \langle x+1,y+1,z+1\rangle $ in the ring of Boolean polynomials. In CoCoA initially I set the ring as ...
0
votes
2answers
29 views

Prove there isn't an isomorphism between quotient polynomial rings

Prove there isn't an isomorphism $$\phi: {{\mathbb Q [x]} \over {I_1}} \to {{\mathbb Q [x]} \over {I_2}}$$ when $I_1=\langle x^2-2\rangle$, $I_2=\langle x^2+2\rangle$. I want to assume there is an ...
1
vote
1answer
12 views

$IM$ not finitely generated , $J \subseteq I$, $JM$ finitely generated; is there some $a\in I$ such that $JM\subsetneq\langle a,J\rangle M$?

Let $R$ be a commutative ring with unity, $M$ be an $R$-module, $I$ be an ideal of $R$ such that $IM$ is not a finitely generated submodule. Let $J \subseteq I$ be a finitely generated ideal such that ...
0
votes
1answer
26 views

$M$ be a finitely generated $R$-module , and $N$ be a submodule of $M$ ; is it possible to have a meaning for $Ann(M)/N$ as an ideal?

Let $M$ be a finitely generated $R$-module, and $N$ be a submodule of $M$; is it possible to have a meaning for $Ann(M)/N$ as an ideal? (I ask this question due to its use in the third line in the ...
0
votes
1answer
31 views

Find a nontrivial proper ideal of $\mathbb{Z}\times\mathbb{Z}$ that is not prime

So I know that $4\mathbb{Z}\times\mathbb{Z}$ is a non-prime ideal of $\mathbb{Z}\times\mathbb{Z}$, and why it is. My question is, how would you find this without testing out many different ideals of ...
1
vote
0answers
27 views

An Ideal in the Group Ring $RG$

So I'm working on Abstract Algrebra (Dummit & Foote). Let $R$ be a commutative ring with identity $1$ and let $G=\{g_1, ..., g_n\}$ be a finite group. Prove that $$I=\{\sum_{i=1}^n ag_i | a\in ...
0
votes
2answers
30 views

Find all prime ideals and maximal ideals of $\mathbb{Z}/12\mathbb{Z}$

How do I go about finding these? I know that the prime and maximal ideals in this case are the same, and that an ideal $M$ is only a maximal ideal of $R$ iff $R/M$ is a field, but I don't really know ...
2
votes
1answer
24 views

Finding a prime in a ring extension using Nakayama's lemma

This is a follow up to my previous question here if $A \subset B$ is a finite ring extension and $P$ is a prime ideal of $A$ show there is a prime ideal $Q$ of $B$ with $Q \cap A = P$. (M. Reid, ...
3
votes
5answers
50 views

Given ring $F[X]/(X^2)$ why is the ideal (X) the unique maximal ideal of the ring [duplicate]

Given ring $F[X]/(X^2)$ I'm trying to understand why the ideal (X) is the unique maximal ideal of the ring. I have figured out that an element in the ring is either in the ideal (X) or is a unit, but ...
2
votes
2answers
53 views

Showing there is a prime in a ring extension using Nakayama's lemma

Here's the problem that I'm working on: if $A \subset B$ is a finite ring extension and $P$ is a prime ideal of $A$ show there is a prime ideal $Q$ of $B$ with $Q \cap A = P$. (M. Reid, ...
-4
votes
1answer
24 views

Subring of a field [closed]

Let $R$ be a subring of a field $F$ such that for each $x\in F$ either $x\in R$ or $x^{-1}\in R$. Prove that if $I$ and $J$ are two ideals of $R$, then either $I\subseteq J$ or $J\subseteq I$.
0
votes
0answers
39 views

nil Jacobson radicals

Let $f$ be an idempotent element in a ring $S$ with Jacobson radical $J$ so that both $fJf$ and $(1-f)J(1-f)$ are nil. I guess that $J$ is nil too, but I am not sure. I know that the former is ...
0
votes
1answer
37 views

Need help giving Abstract Algebra Examples relating to Ideals and Quotient Groups.

I'm having a hard time trying to provide an example of the following two problems: An example of a ring $R$ and an ideal $I$ of $R$ such that neither $R$ nor $I$ has zero divisors but the quotient ...
6
votes
2answers
94 views

Ideal of $\mathbb{C}[x,y]$ not generated by two elements

Consider the ring of polynomials in two variables $\mathbb{C}[x,y]$. Show that the ideal $\langle xy^3, x^2y^2, x^3y\rangle$ cannot be generated by two elements. Until now, I assumed by ...
0
votes
0answers
26 views

Is multiplicative inverse defined for ideal? Eg. $x^3 y\in \langle x^3 y^2\rangle$?

Definition. A subset $I\subset k[x_1,\ldots,x_n]$ is an ideal if i. $0\in I.$ ii. If $f,g\in I$, then $f+g\in I$. iii. If $f\in I$ and $h\in k[x_1,\ldots,x_n]$, then $hf\in I$. ...
2
votes
0answers
20 views

Ideal $K$ in ring of germs is generated by nonnegative functions

I asked a question about details allowing to answer this question earlier today. Unfortunately, I didn't manage to complete the exercise. Since the other questions were about another problem, I write ...
1
vote
2answers
36 views

Ring of germs of smooth functions on $\mathbb{R}^{n}$ in $0$.

First of all, I'm quite new to this theory, so it may be very dumb questions. Sorry for that. Let $R$ be the ring of germs of $C^{\infty}$ functions on $\mathbb{R}^{n}$ in $0$. Let $K$ be the ...
4
votes
1answer
77 views

Is $\mathbb{Z}[\sqrt{15}]$ a UFD?

Let $R=\mathbb{Z}[\sqrt{15}]=\{a+b\sqrt{15}:a,b\in\mathbb{Z}\}$. How do I show that $(3,\sqrt{15})$ is a maximal ideal but not a principal ideal? How do I show that $(3,\sqrt{15})^2$ is a ...
0
votes
1answer
29 views

Showing that $\operatorname{lcm}(a,b)$ is the unique positive generator of $(a) \cap (b)$

Let $a,b \in \mathbb{N}$. $l>0$ is the unique positive generator of the ideal $(a) \cap (b)$. Show that $l = \frac{ab}{d}$ where $d = gcd(a, b)$. I am stuck on this problem. $(a)=\{na: n \in ...
3
votes
1answer
50 views

Principal Ideals and Identity elements

When defining a principal ideal, e.g $I$ $I=aR=(ar:r \in R)$ (an ideal generated by a single element of the ring)do we require the ring to have an identity element and if so, in what manner does ...
1
vote
1answer
29 views

Uncountably many left ideals?

Let $R$ be a following subring of $M_2(\mathbb{C}):$ \begin{equation*} R = \left\{ \begin{bmatrix} a & r \\ 0 & s \end{bmatrix} ~:~ a\in \mathbb{Q} ...
0
votes
1answer
34 views

Determine if an ideal is prime

Let $R=\mathbb{Z}[x,y]$ and $I=(y-x^2)R+(x-4)R$ is ideal $J$ prime ? I tried to produce such $a,b$ that $ab \in J$ but $a,b \not\in J$ but can't find so far
1
vote
1answer
46 views

For ideals $I=(8)$ and $J=(5+5i)$ in $\mathbb{Z}[i]$, what are $IJ$, $I+J$ and $I\cap J$?

Let $I=(8)$ and $J=(5+5i)$ be ideals in $\mathbb{Z}[i]$. How do I find $x,y,z\in\mathbb{Z}[i]$ such that: $IJ=(x)$, $I+J=(y)$, $I\cap J=(z)$? Is it correct that $y=13+5i$ and $x=40+40i$?
-1
votes
1answer
43 views

Prove that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$

In the Milne's book A Primer of Commutative Algebra, pg. 100, there's a proof that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$. I understand the first inequality, ...
1
vote
1answer
15 views

$A_i \subset$ ring $R$ st $\Delta A_i \equiv \{ b-c: b,c \in A_i\} = $ an ideal of $R$, then $\bigcap_i (\Delta A_i) = \Delta (\bigcap_i A_i)$.

Let $A$ be any subset of a ring $R$. $A$ is called a delta generating set of an ideal if $\Delta A = \{b - c: b,c \in A\}$ forms an ideal of $R$. Let $A_i$ be any collection of delta generating sets ...