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
0answers
22 views

Complexity of and an algorithm for finding ideals of a ring?

One of the problems that has been a roadblock in my understanding of ideals has been how one would find them. One way of finding an I of some ring R would be to say $ \forall x \in I, \forall r \in R ...
2
votes
1answer
35 views

ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
1
vote
1answer
31 views

Lie group and generated ideals

I have this question in my textbook, and I can't seem to solve it on my own: Let $P \subset GL(n,\mathbb{C})$ be a subgroup as following: $P$ consists of all matrices in block ...
2
votes
3answers
227 views

Is this example right (ideals of $\mathbb{Z}[x]$)?

I encountered the following problem: Let $I_{0}=\{f(x)\in \mathbb{Z}[x] \ | \ f(0)=0\}$. For any positive integer, show that there exists a sequence of ideals such that $I_0\subsetneq ...
3
votes
1answer
25 views

Ideal in Dedekind domain

Let $D$ be Dedekind domain and $I$ nonempty ideal in $D$. I have to show that there are only finitely many ideals $J$ in $D$ such that $I$ is contained in $J$. My first idea would be: assume that ...
0
votes
0answers
62 views

An $n$-generated ideal of grade $n$ can be generated by an $R$-sequence in any order

It is known to me that if $I$ is an $n$-generated ideal of a commutative ring $R$ with $\operatorname{grade}(I)=n$, then it is generated by an (ordered) $R$-sequence in $I$ of length $n$. I have a ...
0
votes
2answers
62 views

What are the elements like in $\mathbb C[x]/(2x^2+5)$?

As stated, I wonder what are the elements in $\mathbb{C}[x]/(2x^2+5)$ composed like? Since $2x^2+5$ is not monic, it seems to be a little different from the situation like in $\mathbb C[x]/(x^2+5/2)$. ...
4
votes
1answer
50 views

Do $IJ$ and $I\cap J$ coincide if $I$ and $J$ are coprime? Also if ring $R$ has a $1$ and is not commutative?

Let $R$ be a ring (with identity) and let $I,J$ be two coprime (two-sided) ideals in it. In Algebra: Chapter $0$, Aluffi, III. exercise 4.5. the reader is asked to prove that: $$IJ=I\cap J$$ ...
-4
votes
1answer
113 views

An ideal which is not maximal in $\mathbb{C}[x,y,z]$

Show that $$J=(x^2+y^2+z^2+x+y+z, x^5+y^5+z^5+2(x+y+z), x^7+y^7+z^7+3(x+y+z))$$ is not the maximal ideal $m=(x,y,z)$ in $\mathbb{C}[x,y,z]$.
0
votes
0answers
41 views

Universal Basis

How can I prove that $G=\{X-Y^2,XY-X,X^2-X,Y^2-Y^3\}$ is an universal Groebner Basis for the ideal $I=\{X-Y^2,XY-X\}$ in $\mathbb{Q}[X, Y]$ ? any suggestion is good. Thanks.
1
vote
1answer
17 views

Product of ideals is contained in intersection. Seemingly contradiction.

I am reading through a proof where the writer looks at a simple ring $A$ with a nonzero right ideal $M$. At a certain point he comes to this: The poduct ideal $AM$ is a two sided ideal, and so ...
0
votes
1answer
61 views

Intersection of ideals

I am currently studying relations between several kinds of rings and domains. I have seen some properties concerning the sum of ideals : when I and J are finitely generated, then I+J is always ...
0
votes
1answer
24 views

Non irreducible primary ideal in $K[x,y,z]$

this is maybe an easy question but I don't see the answer: I'm trying find a primary ideal in $K[x,y,z]$ non being irreducible (where $K$ is a field) Thank you in advance!
8
votes
3answers
168 views

Polynomial irreducible - maximal ideal

I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal. $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$ $\mathbf 1)$ Is $I_1$ a maximal ideal in ...
0
votes
2answers
31 views

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R-M$ is a unit. Then $R/M$ is a field.

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R$ not in $M$ is a unit. Then $R/M$ is a field. I am solving this question of NBHM 2011. To solve this is ...
0
votes
0answers
39 views

Equivalent definitions of Jacobson rings

We say that a ring $R$ is a Jacobson ring if $$ J(R/I)=\operatorname{nil}(R/I) $$ for every proper ideal $I$ of $R$, where $ J(R)=\bigcap\{M:M \text{ maximal ideal}\}. $ Then it also says, ...
2
votes
1answer
40 views

Computing the Zariski cotangent space

I'm an extreme beginner with algebraic geometry and am trying to get used to things. Say I have some (algebraically closed) field $k$, in $k^2$ I want to compute the Zariski cotangent space, let's say ...
2
votes
1answer
86 views

Example of a Banach algebra $ A $ whose only closed ideals are $ \{ 0 \} $ and $ A $.

I'm trying to come up with an example of a Banach algebra $A$ that is not commutative, unital and such that the only closed ideals are $\{0\}$ and $A$. I already struggled to even come up with a non ...
2
votes
1answer
34 views

Isomophism between rings an two right ideals

Let I, J two right ideals of a ring R such that I+J =R. Show thath the direct sum of I and J is isomorphic to the direct sum of R and the intersection of I and J. Can anyone please give me at least ...
0
votes
2answers
36 views

Not so usual equivalence of maximal left ideal of a ring

I was reading Foundations of Module and Ring Theory and i found this equivalence of maximal left ideal as exercise in the the first chapter: A left ideal $I$ of a ring $R$ is a maximal if and only ...
2
votes
1answer
35 views

Ideals in a Quadratic Number Field

Show the ideal $I=\langle4,2+2\sqrt{-29}\rangle$ in $\mathbb{Z}[\sqrt{-29}]$ satisfies the equality $\langle8\rangle=I^{2}$ of ideals in $\mathbb{Z}[\sqrt{-29}]$. I tried to factorise $x^{2}+29$ over ...
1
vote
3answers
57 views

$a^n = 0 \implies a \in P$ (where $P$ is a prime ideal)

Is the above true? (I think it is!) if so, please can somebody explain why? I don't see it!
1
vote
1answer
42 views

Annihilator of a quotient module

Let $J$ be an ideal of $R$, and $M$ a right $R$-module. Since $Jr \subseteq J$, $M / MJ$ is naturally a right $R$-module. Since it seems relevant to another problem, I am trying to determine ...
3
votes
0answers
37 views

When are all (prime) ideals of an $R$-algebra, extensions of (prime) ideals of $R$?

Let $f:R\rightarrow R'$ be a homomorphism of commutative noetherian rings. When are all (prime) ideals of $R'$ extensions of (prime) ideals of $R$? Is it true for the case $R'$ is $R$-flat?
0
votes
1answer
29 views

principal ideals, integral domains, ideals,?

I am stuck trying to grasp this concept. I know that $\Bbb{Z}$ is a PID, $R=\Bbb{Z}[X]$ is not a PID, $\Bbb{Z}[i]$ is a PID. If someone could help me grasp these concepts it would be helpful. ...
0
votes
2answers
22 views

Finding a ring isomorphism

Let $\phi : R \to R'$ be a ring epimorphism and $J\lhd R'$ an ideal of $R'$. Indicate a ring isomorphism $\psi: R/\phi^{-1}(J) \to R'/J$ The only thing i know about this problem is that ...
0
votes
1answer
14 views

About the connection between ideals and homomorphisms

I know that for a homomorphism of rings $\psi : R\rightarrow S$ we have that $\ker\psi$ is an ideal of $R$. I was wondering if the opposite direction is true: Let $I$ be an ideal of $R$. Then does ...
0
votes
1answer
29 views

How to find the ideals of $\Bbb{Z}_n$

I have a homework problem to find the maximal ideals in $\Bbb{Z}_8$, $\Bbb{Z}_{10}$, $\Bbb{Z}_{12}$, and $\Bbb{Z}_n$. That question has already been asked on here, but I don't even understand how to ...
2
votes
1answer
40 views

Ideals generated for commutative ring

The following is a problem from the Gallian book. I'm trying to understand what exactly this ideal is and how to verify that it is in fact an ideal. "Let $R$ be a commutative ring with unity and let ...
3
votes
2answers
60 views

How can I prove an ideal is a product of two irreducible ones

I'm trying to solve this question: I have a guess that $(6+\sqrt{11})=(2,4+\sqrt{11})(2,-3\sqrt{11})$ using some formulas in this book page 48. However I couldn't verify if the multiplication of ...
0
votes
0answers
61 views

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$?

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$ ? $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)=\{ (0),\ (\tilde{x}-a,\tilde{y}-b),\ b^2=a^3\}$.
0
votes
1answer
23 views

Factorisaing ideals in quadratic number fields

Show there is an ideal $a$ in $\mathbb{Z}[\sqrt-29]$ satisfying the equality $\langle8\rangle=a^{2}$. I tried to factorise the minimal polynomial over $\mathbb{F}_{8}$ but it does'nt seem to work, ...
2
votes
0answers
31 views

Dedekind's criterion clarification

Dedekind's criterion gives a way of factoring $p\mathcal{O}_K$ into prime ideals. (See http://math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf) Is it true that the prime ideals ...
1
vote
1answer
47 views

Symmetric powers of ideal quotients in a local ring.

Let $R$ be a local ring and $I \subset R$ any ideal. When is it the case that $(I \: \backslash I^2)^n = I^n \: \backslash I^{n+1}$? Put another way, when is the natural map $\text{Sym}^n(I/I^2) ...
3
votes
3answers
52 views

2-sided ideal in $\mathfrak{M}_{2\times 2}(\mathbb{Z})$ ring

Hi there: I know $\mathfrak{M}_{2\times 2}(\mathbb{Z})$ has left and right ideal, but is it true it does not have 2-sided ideal? If there is, could you give me an example. Thank you very much.
3
votes
4answers
130 views

Finding generators for an ideal of $\Bbb{Z}[x]$

We know that $\Bbb{Z}$ is Noetherian. Hence, we can conclude that $\Bbb{Z}[x]$ is Noetherian, too. Consider the ideal generated by $\langle 2x^2+2,3x^3+3,5x^5+5,…,px^p+p,…\rangle$ for all prime ...
2
votes
1answer
33 views

Principal ideals as generated groups

This seems like a pretty simplistic question, but I can't find a solid, non-ambiguous answer to it. The question I'm given: Is $I$ a principal ideal of $R$? Given: $R=\mathbb{Z}$ and $I=\left\langle ...
0
votes
0answers
41 views

How to decide if an ideal in $\mathbf Q[X,Y]/(P)$ is principal?

Let $P(X,Y)$ be an irreducible polynomial in $\mathbf Q[X,Y]$. Given an ideal $I$ of the quotient ring $\mathbf Q[X,Y]/(P)$ (say given by a set of generators) how can I decide if $I$ is principal or ...
0
votes
1answer
27 views

Let $f(x)\in F[x]$ ($F$ a field) be irreducible and let $\alpha$ be a root of $f(x)$. Then $h(x)\in(f(x))\Leftrightarrow h(\alpha)=0$?

Let $F$ be a field and $f(x)$ an irreducible polynomial in $F[x]$ such that $\alpha$ is a root of it: $f(\alpha)=0$. Now, let $(f(x))\subset F[x]$ denote the ideal generated by $f(x)$. My question is: ...
1
vote
1answer
38 views

Krull dimension, commutative algebra. Eisenbud, Exercise 10.3

This is the exercise. Let $k$ be a field. Prove that $k[x]\times k[x]$ contains a principal ideal of codimension $1$, although it's not a domain. Now, I have to find a principal ideal prime, such ...
0
votes
0answers
31 views

Factorisation in rings of algebraic integers

Determine the prime factorisation of the principal ideal $\langle15\rangle$ in $\mathbb{Z}[\sqrt{-14}]$. Can I use the fact $\langle15\rangle=\langle3\rangle\langle5\rangle$ and the factorisation ...
2
votes
1answer
42 views

Isomorphism between a sub-ring to $\Bbb Q$, the rational field

Let $R$ be a commutative ring with an identity, which contains $\Bbb Z$, the integer field, as a sub-ring with the same identity element. Let $I$ be a maximal ideal in $R$. I need to prove that if ...
0
votes
1answer
51 views

Addition and Multiplication table for Ring/Ideal

I'm not sure if it's possible to show it here, but how would the addition and multiplication table look like for R/I (where R is rings with ideal I) when $$ R = Z_{12} \text{ and } I = \{0,3,6,9\} ...
2
votes
0answers
29 views

Lattice with $3$ operations.

If $R$ is a commutative ring and $\mathcal I(R)$ denotes its set of ideals then I know that $\mathcal I(R)$ can be looked at as a complete lattice with intersection $I\cap J$ and addition $I+J$ as ...
1
vote
1answer
59 views

A theorem about ideals of $K[T_1,\ldots,T_n]$ and their generators

Suppose that $L\subseteq K$ is a field extension ( we are in characteristic $0$) and moreover that $\mathfrak a\subseteq K[T_1,\ldots,T_n]$ is an ideal ($T_1,\ldots,T_n$ are indeterminates). I have ...
0
votes
2answers
31 views

Product of two ideals in Dedekind domains

Let $\mathcal{O}$ be a Dedekind domain and $I=(x_1,\ldots, x_n),J=(y_1,\ldots, y_m)\subseteq \mathcal{O}$ two ideals. Is it possible, that $IJ\neq K$ with $K$ the ideal generated by the products ...
4
votes
2answers
73 views

Computing the radical of an ideal

What is the best way to compute $\sqrt{(X^2-YZ,X(1-Z))}$ ? This is after using Nullstellensatz by the way as I thought it would be easier to compute a radical than finding the vanishing ideal.
1
vote
0answers
27 views

A problem involving ideals and prime ideals. [duplicate]

Please help me with a solution to this problem. Let $R$ be a commutative ring. Let $A_1, A_2$ be two ideals of $R$, and $P_1, P_2$ two prime ideals of $R$. Assume that $A_1 \cap A_2 \subseteq P_1 ...
2
votes
2answers
54 views

How to show that $(Y- X^2, Z - X^3) \subseteq k[X,Y,Z]$ is a prime ideal?

I suppose that $k$ is an algebraically closed field (actually, my goal is to show $\mathcal{I}(\mathcal{V}(Y- X^2, Z - X^3)) = (Y- X^2, Z - X^3)$). (But I think algebraically closed is not necessary ...
3
votes
1answer
74 views

Prove that $m^2$ is primary

Let $m$ be a maximal ideal. I'm having a hard time proving that $m^2$ is primary. Let ${xy\in m^2}$ so $xy=t_{1}s_{1}+...+t_{n}s_{n}$ where the $t_{i},s_{i}$ are in $m$.