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
1answer
31 views

When a given ideal is a radical ideal

I am wondering if there are any canonical methods for checking whether a given ideal is radical. For example, I got stuck on the following example: Let $f=x+2y-z$ and $g=z-2w$ and let $I$ and $J$ be ...
1
vote
1answer
59 views

Characterizing maximal ideals in $\mathbb{Z}[x]$

I need to prove this: Let $I\subset\mathbb{Z}$ be the ideal generated by $\{p,f(x)\}$, with $p$ prime in $\mathbb{Z}$. Then $I$ is maximal iff $f(x)$ is irreducible modulo $p$. So I was trying to ...
1
vote
1answer
48 views

Characterizing Prime and Maximal Ideals in a nice Ring

Consider the "nice" ring $(\mathbb{Z}/20\mathbb{Z})[x]$ and I am trying to list all the prime and maximal ideals of this. The reason I call this a nice (or manageable) ring is because we ...
3
votes
1answer
45 views

Show that the trace class operators on a Hilbert space form an ideal

Let $(H, (\cdot, \cdot))$ be a separable Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $\{\phi_n\}_{n=1}^\infty$ is an orthonormal basis for $H$. Let $\mathcal{B}(H)$ ...
2
votes
0answers
281 views

Multiplicity and regular sequences

We define multiplicity of a module $M$ of dimension $d>0$ as $$e(M) := \operatorname{lc} (P_M) (d-1)!,$$ where $P_M$ denotes the Hilbert polynomial of $M$ and $\operatorname{lc}(P_M)$ its leading ...
2
votes
2answers
35 views

Nil radical of an ideal on a commutative ring

This is a problem of an exercise list: Let $J$ be an ideal of a commutative ring A. Show that $N(N(J))=N(J)$, where $N(J)=\{a \in A; a^n \in J$ for some $n \in \mathbb{N}\}$. What I did: ...
0
votes
0answers
47 views

Polynomial Ideals and Transverse modules

I have a given ideal and I want to find the "smallest" ideal so that when I add it to the original one, I get another certain ideal. Let $\mathcal{E}$ be the ring of smooth function germs ...
1
vote
1answer
126 views

Writing $I= (xz-y^2, yt- z^2)$ as an intersection of prime ideals

I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea? For the moment, I've noticed that $I$ is radical, then it suffices to ...
0
votes
1answer
60 views

Irreducible ideals are prime in polynomial rings

Let $k$ be an algebraically closed field and $R$ the polynomial ring in $n$ variables over $k$. If $J$ is an irreducible ideal of $R$ then it is a prime ideal as well. To establish this statement ...
2
votes
1answer
44 views

An easy looking quotient of a local ring

$k$ is a number field, $R$ its ring of integers and $\mathfrak p$ a nonzero prime ideal of $R$. Let $R_\mathfrak p$ be the localization of $R$ at $\mathfrak p$. Is it true that $R_\mathfrak ...
1
vote
3answers
74 views

why is maximal ideal of a ring of integers generated by a single prime number?

I cannot understand why maximal ideal in a ring of integers is generated by a single prime number. For example, if we choose 2 and 3 as generators of ideal, then all multiples of 2 and 3 will be in ...
0
votes
0answers
26 views

For the ring of natural numbers, what happens if we convert all $k>1$th power of prime numbers to be zero?

So there is a ring of natural numbers. Now someone decides to make all $k>1$th power of prime numbers to be zero. So $2^2 = 2^3 = 2^4 = ... = 3^2 = 3^3 = ... = 0$. Or we can say that these elements ...
1
vote
3answers
87 views

What exactly is a maximal ideal?

I am confused about the definition of maximal ideal. Suppose that there is ring $R$. Now if we select the whole $R$ to be an ideal, then wouldn't this be maximal ideal? Or is the definition of maximal ...
0
votes
1answer
26 views

Image of a Prime ideals in quotient ring

I am having confusion regarding the relationship between the prime ideals of a Ring and a certain quotient of a Ring. Let A be a commutative ring with identity, and a an ideal of A. Clearing there is ...
2
votes
2answers
43 views

Is the ideal $(2,x^4+x^2+1)<\mathbb{Z}[x]$ maximal?, principal?

I'm trying to solve the following problem: Let I=$(2,x^4+x^2+1)<\mathbb{Z}[x]$ be an ideal. Is $I$ maximal? Is $I$ principal? Any help would be appreciated.
2
votes
0answers
22 views

Prove that $\mathfrak{p}$ is totally split in $L/K$ and $L'/K$ $\Rightarrow$ totaly split in $LL'/K$

Assume that $K$ be a number field and $L/K$, $L'/K$ are two separable extensions. Now let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$. Then if $\mathfrak{p}$ is totally split ind $L$ and $L'$, ...
5
votes
1answer
72 views

$(x,y)$-primary ideals

I want to find all ideals $I$ in $\mathbf{C}[x,y]$ with $\sqrt{I}=(x,y)$ and $\dim_{\mathbf{C}}\mathbf{C}[x,y]/I=2$. I have no clue how to about it, I mean I can write down some examples, ...
2
votes
2answers
43 views

For which values of $a\in\mathbb ℤ/3\mathbb ℤ$ is the quotient $\mathbb ℤ/3\mathbb ℤ[x]/(x^3+x^2+ax+1)$ a field?

I'm trying to solve the following problem: Determine for which values of $a\in\mathbb{Z}/3\mathbb{Z}$ the quotient $Q_a=(\mathbb{Z}/3\mathbb{Z})[x]/(x^3+x^2+ax+1)$ is a field. I see two options: ...
2
votes
3answers
129 views

Isomorphic quotient of a module over Noetherian commutative ring

I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
2
votes
2answers
93 views

Finding a maximal ideal and a prime ideal in $\mathbb Z_8[x]$

$1.$ Find a maximal ideal and a prime ideal in $\mathbb Z_8[x]$ Attempt: Finding a maximal ideal, I am not sure how do I go about it. $\mathbb Z_8[x]$ is not a $PID$, so there's no use finding ...
7
votes
1answer
58 views

An extension of an algebraic number field which makes an integral ideal $I$, a principal ideal

I want to show that, given an ideal $I \subseteq \mathcal O_K$ (where $K/\mathbb Q$ is an algebraic number field), there is a finite extension $K'/K$ such that, $I\mathcal O_{K'}$ becomes a principal ...
0
votes
1answer
36 views

Does the relation $\pi(S_{i})=S^{-1}R-P_{i}\cdot S^{-1}R$ hold for prime ideals $P_i$ in a commutative ring $R$?

Let $R$ be a commutative ring. Let $P_{i}$, $1\leq i\leq n$ be prime ideals none of which are contained in each other. Let $S=R-(\cup_{i=1}^{n} P_{i})$. Then $S$ is a multiplicatively closed set and ...
2
votes
1answer
89 views

In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal

Let $k$ be a field and let $A \subset B$ be two finitely generated $k$-algebras. Prove that the contraction of any maximal ideal of $B$ is a maximal ideal of $A$. thank you very much again!
2
votes
3answers
52 views

Example of ring $R$ with ideals $I\neq J$ such that $R/I \cong R/J$ as modules

It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample? I believe I've found an answer, ...
1
vote
1answer
24 views

Equivalent definitions of fractional ideals

Let $R$ be an integral domain and $K$ its field of fractions. The usual definition of fractional ideal $I$ ($I$ is an $R$-submodule of $K$) is that for some nonzero $r\in R$ we have $rI\subset R$, and ...
1
vote
1answer
36 views

Lie ideals of $gl_n(K)$

I am looking for some reference where I can find a detailed study of the Lie ideals of the general linear Lie algebra $gl_n(K)$ with the bracket $[A,B]=AB-BA$, where $K$ is a field (if there are ...
3
votes
2answers
125 views

If for any two principal ideals one contains another, then for any two ideals one ideal contains another

Let $R$ be a commutative ring with identity. Assume that for any two principal ideals $Ra$ and $Rb$ we have either $Ra\subseteq Rb$ or $Rb\subseteq Ra$. Show that for any two ideals $I$ and $J$ in ...
1
vote
1answer
24 views

ideal,ring,flat module,modules over R

Is there a characterization of modules (AND equivalent characterizations of rings R) over integral domains R with the property that each left ideal in R is flat?When all left ideals are ...
2
votes
2answers
43 views

Finding ideal representatives in the class group of $\mathbb{Q}(\zeta_{23})$

I know that $\mathbb{Q}(\zeta_{23})$ has class number 3, and I am wondering how I can find ideal representatives of the two nonprincipal classes in the class group. I have tried looking at examples ...
1
vote
1answer
22 views

Expressing a hereditary subalgebra in terms of a state

Let $\mathcal{A}$ be a C*-algebra and $\phi$ a state on $\mathcal{A}$. Then, it's not hard to see that $\mathcal{L} = \left\{ x : \phi(x^*x)=0 \right\}$ is a closed left ideal of $A$ and so ...
1
vote
1answer
67 views

The geometric interpretation for extension of ideals?

Suppose $f\colon B\to A$ is a ring homomorphism, and $I\subseteq B$ is an ideal. What's the geometric interpretation for the extension $f(I)A$ of the ideal $I$? Especially, I'm interested in the case ...
1
vote
2answers
102 views

Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?

Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is a positive integer $n > 1$ such that $a^n =a$. Prove that every prime ideal of $R$ is a maximal ideal of R. ...
2
votes
3answers
72 views

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal Attempt: Let $R$ be the principal ideal domain. A principal ideal domain $R$ is an integral domain in which ...
0
votes
1answer
41 views

Suppose that $I$ is an ideal of $J$ and that $J$ is an ideal of $R$. prove that if $I$ has a unity then $I$ is an ideal of $R$

Suppose that $I$ is an ideal of $J$ and that $J$ is an ideal of $R$. prove that if $I$ has a unity then $I$ is an ideal of $R$ Attempt: Given that $I$ is an ideal of $J$ which means : ...
2
votes
1answer
43 views

Prime radical that is nil but not nilpotent

Please help me to show that the prime radical of the ring $R=\prod\limits_{n = 1}^\infty { \mathbb{Z} /2^n\mathbb{Z} } $ is nil but not nilpotent.
0
votes
1answer
81 views

The nil-radical is an intersection of all prime ideals proof

Every proof I found online made the same implications. Take one for example: http://www.artofproblemsolving.com/Wiki/index.php/Nilradical I'm quoting the relevant part, which confuses me: "To show ...
3
votes
0answers
122 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
-6
votes
2answers
301 views

Quotient ring isomorphism

I think that if $A$ is any commutative ring with unity and $q\in A$, $p\in A[x]$ then we have $A[x]/(q,p)\cong A/(q)[x]/(\bar{p})$ where $\bar{p}$ denotes the class of $p$ in $A/(q)[x]$. Is this true? ...
-2
votes
1answer
61 views

Given ring and ideal, How to prove that the intersection of ideals is an ideal

Given $R$ is a ring, $X\subseteq H_i$ and $H_i$ is an ideal of $G$ for each $i=1,2,...,n$. Prove that $H_1∩H_2∩...∩H_n$ is an ideal of $G$ and contain $X$. That is a question I get from random ...
4
votes
1answer
42 views

Quotient $M/M^2$ is finite dimensional over $R/M$ in local Noetherian ring?

I have that $R$ is a Noetherian local ring with maximal ideal $M$, and I want to show that $M/M^2$ is a finite dimensional vector space over the field $R/M$. I think I've proved this (though I ...
4
votes
1answer
57 views

What kind of rings have exactly three ideals?

What kind of rings(commutative, w/ unity) have exactly three ideals? I know that those with exactly two ideals are "the fields", but what about three? Is there a fancy name for them?
0
votes
1answer
21 views

Finding a particular principal open subset of $Spec R$

Let $V\subseteq U$ be open subsets of $X=\text{Spec } R$, where $R$ is a commutative ring. So $V$ is the set of prime ideals not containing some ideal $I$, and $U$ is the set of prime ideals not ...
2
votes
1answer
59 views

Locally unital ideals [duplicate]

Let $R$ be a ring with unity not necessarily commutative and $I$ an ideal of $R$. Let for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$. Note that $c$ is related to $a$. ...
0
votes
0answers
42 views

Prime ideal is contraction of prime ideal iff it's saturated

Let $\varphi: A\to B$ be a commutative ring homomorphism and $P$ a prime ideal of $A$. The expansion of an ideal $I\subset A$ is the ideal generated by $\varphi(I)$ in $B$, and the contraction of an ...
1
vote
1answer
39 views

Which is a subring? Which is an ideal?

We are having ring $\mathbb{Z}[\sqrt{-6}]$. Which of the sets is subrings of $\mathbb{Z}[\sqrt{-6}]$ and which are ideals? $\mathbb{Z}+5\mathbb{Z}[\sqrt{-6}]$ $5\mathbb{Z}+\mathbb{Z}[\sqrt{-6}]$ ...
0
votes
2answers
57 views

Maximal ideal which isn't principal

Let $J=(x-2,x-y^2-3)$ ideal in the polynomial ring $\Bbb R[x,y]$. Please help me prove that $J$ is a maximal ideal which isn't principal, and that $\Bbb R[x,y]/J \cong \Bbb C$.
0
votes
1answer
27 views

Specific Ideal determinations for a Ring

Uploaded in a picture, rather than typing it all. Note that these are unmarked questions from a sample exam. Just trying to study, but have forgotten almost everything (three major exams before this ...
1
vote
2answers
71 views

A question about the Zariski Topology

Let $\{a_i\}$ be an infinite set of ideals in commutative ring $R$. Is $\bigcap\limits_{i=1}^\infty a_i$ not defined? I am trying to understand Zariski Topology. Here, $V(\bigcap a_i)= ...
0
votes
1answer
34 views

A question about ideals.

Let $A$ and $B$ be arbitrary subsets of a ring. Then $V(A\cup B)=V(A)\cap V(B)$. Here, $V(X)$ is the set of prime ideals containing $X$. Let $W(X)$ be the set of ideals (any sort of ideals) ...
1
vote
1answer
18 views

Determining if any of these three are an ideal of $\mathbb{R}[x]$

$\mathbb{R}[x]$ denotes the ring of polynomials in $x$ with real coefficients. Let $I \subset \mathbb{R}[x]$ be the subset of those polynomials with constant coefficient $0$, and let $J \subset ...