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, ...

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What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of $...
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194 views

Ideals in $C[0,1]$ which are not finitely generated (From Atiyah- Macdonald )

I'm trying to solve the following problem from Atiyah-Macdonald: Is the ring of continuous function on $[0,1]$ is Noetherian ? Certainly not, here are two non terminating ascending chain of ...
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329 views

Maximal ideal space of $C^*$-algebra of Riemann integrable functions

Let $R([0,1])$ be the unital commutative $C^*$-algebra of complex valued Riemann integrable functions on $[0,1]$ with pointwise operations and the supremum norm. In the 1980 paper The Gelfand space ...
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222 views

Converse of Chinese Remainder Theorem

Chinese Remainder Theorem for commutative rings with identity Let $R$ be a commutative ring with identity. If $I, J$ are ideals of $R$ satisfying $I+J=R$, then there is an isomorphism of rings: $$R/(...
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Looking for a special class of ideals such that If every ascending chain of ideals from this class stabilizes, then $R$ is a Noetherian ring.

A commutative ring $R$ is called Noetherian if any one of the following holds: $1.$ Every ascending chain of ideals in $R$ stabilizes, that is, $$ I_1\subseteq I_2\subseteq I_3\subseteq\cdots $$ ...
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158 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
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104 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
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305 views

Showing an ideal is prime in polynomial ring

Let $k=\mathbb{C}$ and let $J$ the ideal $(xw-yz,y^{3}-x^{2}z,z^{3}-yw^{2},y^{2}w-xz^{2})$. I want to see why $J$ is a prime ideal in $k[x,y,z,w]$. I know that $Z(J)$ (the zero set of $J$) is ...
5
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57 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net $\{e_\alpha\}\...
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41 views

Stuggling to understand ideal powers

In my current algebraic number theory course we have defined the multiplication of 2 ideals as the smallest ideal containing all products of elements of both, [i.e: let I and J be ideals of a ring R,...
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201 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 $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; $I\cap(J+K)=(I\...
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250 views

Irreducible homogeneous ideals

I have the following question: Let $I$ be a homogeneous ideal. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideals? So, is it ...
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131 views

Counterexamples to the avoidance lemma for arbitrary ideals

Let $A$ be a commutative ring with $1$. Let $I$ and $J_k$, $k=1,\dots,n$ be ideals of $A$ with $I\subseteq \cup _{k=1}^n J_k$. Then I have obtained the following: (1) If $J_k$, $k=1,\dots,n$, are ...
5
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271 views

Ideals in Gaussian integers

Let $R:=\mathbb{Z}[i]$. Prove that every nonzero prime ideal $\mathfrak{P}$ of $R$ belongs to one of the following families: 1) $\mathfrak{P}=(1+i)R$ 2) $\mathfrak{P}=(a+bi)R$ where $a,b\in\mathbb{Z}...
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220 views

Non-cyclic unit groups

Is there any way to motivate why certain factor rings of $\mathbb{Z}, \mathbb{Z}[i]$, etc., to a prime power have non-cyclic unit groups? For example, the only such non-cyclic unit groups of factor ...
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43 views

Left ideals of $M_2(K)$ with $K$ a field

Is it true that the only proper left ideals of $M_2(K)$, the ring of the matrices whose coefficients are in a field $K$, are $$ \left\{\begin{pmatrix}ah & ak \\ bh & bk\end{pmatrix}: a,b \in K\...
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51 views

Maximal ideal containing functions with compact support

I recently proved the following statement: Let $M$ be a smooth manifold and let $I \subseteq C^\infty(M)$ be an ideal such that $C^\infty(M)/I \cong \mathbb{R}$ (such an ideal is clearly maximal, ...
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32 views

Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in $E^{**}$...
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65 views

Checking that a set is a finitely generated ideal

The exercise asks us to prove that $I = \{ f \in \Bbb R[X,Y,Z] \mid f(a,b,c) = 0, ~\forall\,(a,b,c)\in \Bbb S^2 \}$ is a finitely generated ideal of $\Bbb R[X,Y,Z]$. Well, clearly $I$ is an ideal of $...
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337 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 ...
4
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88 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
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79 views

Isomorphism of K-algebras

Let $k[x]$ be a polynomial ring and $I$ an ideal (resp. graded). If $k\subset K$ is a field extension, then prove that there is a natural (resp. graded) isomorphism of $K$-algebras: $$K[x]/IK[x]\cong ...
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290 views

Primary decomposition of ideals

How to find a primary decomposition of the ideal $I = (X^2, XY, XZ, YZ)$ in the ring $k[X,Y,Z]$? Is there a general rule for finding primary decompositions? Also how to show that $(X,Y)^{308}$ is a ...
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41 views

How are varieties related polynomials?

My teacher says that varieties and ideals are related to each other while I tend to mix polynomials and varieties in my terminology. Could some explain how varieties are related to polynomials? And ...
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$R=\{ m+nr\sqrt{2} \mid m,n \in \Bbb Z \}$ and $I_{a,b}=\{ ma+n(b+r\sqrt{2}) \mid m,n \in \Bbb Z \}$

Let $r$ be a natural number and $R=\{ m+nr\sqrt{2} \mid m,n \in \Bbb Z \}$. We can show that $R$ is a subring of the ring $\Bbb Q [\sqrt{2}]$. My questions are as follows: $(1)$ Suppose that a ...
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43 views

How else can I tell I can do this with $5$ but not $2$ or $3$ in $\textbf{Z}[\sqrt{30}]$?

In $\textbf{Z}[\sqrt{30}]$, the number $5$ splits, since, for example, $N(5 + \sqrt{30}) = -5$. But the ideal $\langle 5 \rangle$ is a ramifying ideal, since it is equal to $\langle 5, \sqrt{30} \...
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39 views

Radical of an Ideal Proof

$\sqrt{\sqrt{\mathfrak{a}}} = \sqrt{\mathfrak{a}}$ $\sqrt{\sqrt{\mathfrak{a}}} \subseteq \sqrt{\mathfrak{a}}:$ Let $x \in \sqrt{\sqrt{\mathfrak{a}}}$, then $x^n \in \sqrt{\mathfrak{a}}$ for some $...
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48 views

Intersection $I \cap \mathbb{Z}$ where $I$ is an ideal of $\mathbb{Z}[X]$

is there a reasonable algorithm that allows, given finitely many generators of an ideal $I$ of $\mathbb{Z}[X]$, to find the intersection $I \cap \mathbb{Z}$? Thank you.
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50 views

Proof verification: $\langle 2, x \rangle$ is a prime, not principal ideal

I want to prove that $\langle 2, x \rangle$ is a prime, but not principal, ideal of $\mathbb{Z}[X]$. $\langle 2, x \rangle$ is prime: Suppose $f(x) = f_n x^n + \ldots + f_1 x + f_0$ and $g(x) = g_m x^...
3
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62 views

$\mathbb{Z}[i]$ is principal. And what are the units

I have elements of the form $a+bi$. I have attempted to consider arbitrary ideals in $\mathbb{Z}[i]$. If $N$ is ideal and $N=\{0\}$ then it is generated by $0$. If $N$ is not trivial, then exists $f=...
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Please help in proving $ab=0 \space \forall a,b$ in a ring $R$ where the only right ideals of $R$ are the trivial ones and $R$ is not a division ring.

I have the following question I am trying to solve: Let $R$ be a ring such that the only right ideals of $R$ are $(0)$ and $R$. Prove that either $R$ is a division ring or that $R$ is a ring with ...
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33 views

Unique prime ideal containing $(2)$ in $\mathbb{Z}[\sqrt{-3}]$

I'm having trouble with an algebraic number theory problem. Let $R = \mathbb{Z}[\sqrt{-3}]$. The problem is to show that $(2, 1 + \sqrt{-3})$ is the unique prime ideal containing the ideal $(2)$, and ...
3
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110 views

A Problem for Nil-Ideals

Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for non-commutative rings I think this may not be true. ...
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Module endomorphisms with the same kernel

Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in R[x_{...
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63 views

Left ideals in a subring of $M_2(R)$

Let $R$ be a 2-dimensional complete regular local ring $R$ over an algebraically closed field $k$, that is $R\cong k[[x,y]]$. Now look at the the following subring $A$ of $M_2(R)$: $A=\begin{pmatrix} ...
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63 views

Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
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78 views

Is there a classification of ideals of $\mathcal O_K$ ($K$ quadratic) over ramified and split primes depending on $d \pmod 4$?

I am unsure if the following argument is correct. I have not seen something like this in my course, so I'm a bit skeptical, since this seems like a very simple way of computing norms of ideals. If ...
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44 views

Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal.

Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal. Hint: Give a counter-example This is my rough proof to this question. I was wondering if anybody can look over it and ...
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138 views

Can a local ring have more than one prime ideal?

A local ring is defined as a ring which has a unique maximal ideal. This unique maximal ideal consists of only non-units and contains all the non-units of the ring $R$. So examples of local rings ...
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Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators $f_{1},......
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About quotient ring

I want to find the value $|\mathbb{Z}({\sqrt{2})/(3+\sqrt{2})}|,|\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}|$ and also the number of ideals of $\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}$. But still not ...
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A question about the consequence of Prime Avoidance.

I have found the following statement: Let $R$ be a Noetherian ring and $x$ is a non-zero divisor of $R$. Let $P$ be a prime ideal associated to $xR$. Then by Prime Avoidance there exists a non-zero ...
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Proof about affine varieties

Ok so I have that $k$ is algebraically closed and $F$ is an element of $k^n$, and it is a reduced polynomial. We have that $V = V(F)$. In the book it says prove that $F$ generates $I(V)$ but in my ...
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306 views

Proof of going-up theorem

Can you tell me if my proof is correct? Thanks. (I'm using propositions 5.6 and 5.10 from Atiyah-Macdonald which I proved separately.) Theorem: Let $R$ be integral over $S$. Let $p_1 \subset p_2$ be ...
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40 views

$\mathbb{Z}[\sqrt{10}]$ is noetherian

How can we prove that $\mathbb{Z}[\sqrt{10}]$ is noetherian except by using Hilbert basis theorem? How can we find a sequence of ideals that satisfy the ACC?
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About the discriminant ideal

Let $E/K$ be a separable field extension of degree $n$, let $A$ be a Dedekind Domain which quotient field is $K$, and let $B$ be the integral closure of $A$ in $E$. Then we have that the ideal ...
2
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46 views

Existence of minimal prime ideal contained in given prime ideal and containing a given subset

Let $R$ be a unital commutative ring, $P$ $\subseteq$ $R$ a prime ideal, $X\subseteq P$ a subset. Show there exists a minimal (inclusion minimal) prime ideal contained in $P$ which contains $X$. My ...
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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 ...
2
votes
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68 views

$J$ maximal ideal of $A$ $\iff $ $A/J$ is field

I'm wondering where did this complicated proof that $J$ is a maximal ideal $\iff$ $A/J$ is a field. Is there an easy to look case where we can clearly see that when we take the quotient of the ring ...
2
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67 views

Trying to understand why the associated primes of some ideal are minimal.

In a comment to this answer it is mentioned that all the associated primes of $(XY,(X-Y)Z)$ are minimal using some Cohen-Macaulay ring. Since I don't know anything about Cohen-Macaulay rings, is ...