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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Hilbert-Burch theorem characterizes perfect ideals of grade $2$

Bruns and Herzog in their book Cohen-Macaulay Rings, page 120 write: "The Hilbert-Burch theorem 1.4.17 identifies perfect ideals of grade $2$ as the ideals of maximal minors of certain matrices". ...
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Height of a specific maximal ideal

Let $k$ be a field, $k[x,y^2,xy,y^3]$ our ring and $\mathfrak a$ the ideal generated by $x,y^2, xy,y^3$. I want to determine the height $h(\mathfrak a)$ of $\mathfrak a$. My ideas: We see easily ...
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Is every left maximal ideal the annihilator of a simple left module?

In my version of Noncommutative Algebra, by Benson Farb & R. Keith Dennis, in chapter I, section 2 on the Jacobson radical, it is claimed that … each maximal left ideal $I$ is the annihilator ...
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Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
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Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
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Let $k$ be a division ring, then the ring of upper triangular matrixes over $k$ is hereditary

I'm reading Ring Theory by Louis H. Rowen, and he claimed that The ring of upper triangular matrices over a division ring is hereditary (it's on page 196, Example 2.8.13 of the book). I think it ...
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$\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = \{ \mathfrak{p}A_\mathfrak{p}\} $

Let $k$ be a field, $A = k[X_1,X_2,...]$, $\mathfrak{p} = (X_1,X_2,...)$, $I = (X_1^2-X_1,X_2^2-X_2,...)$, $M= A/I$. I am trying to show that $\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = ...
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Is every module a direct limit of cyclic modules?

I want to show that $M$ is $A$-flat is equivalent to $Tor_1^A(M,A/I)=0$ for every finitely generated ideal $I$. I want to show $Tor^A_1(M,N)=0$ for any $A$-module $N$. Is every module a direct ...
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Maximal ideals in the ring of eventually constant sequences of real numbers

For homework I am studying the ring $R$ of eventually constant sequences of real numbers (with multiplication and addition defined componentwise). What are the maximal ideals of $R$? By looking at ...
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Non-radical ideal giving the empty set.

Let $R$ be the polynomial ring of $n$ variables over $\mathbb C$. It is known that a radical ideal $I (\ne R)$ defines a non-empty set $\mathbf V(I) \subset \mathbb C^n$. I am looking for a ...
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Polynomial rings over a field and maximal/prime ideals

Let $F$ be a field , I want to prove that every proper nontrivial prime ideal of $F[x]$ is maximal. My definitions of prime/maximal ideals are as follows: $N$ is a prime ideal of $R$ iff $ab \in N ...
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60 views

A non-zero and non-invertible element in a noetherian integral domain has a decomposition into irreducible elements

Let $R$ be a noetherian integral domain. I want to show that any non-zero and non-invertible element $a$ can be written as a finite product of irreducible elements. my ideas: I should argue by ...
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$\dim (A/I) \le \dim (A)$

Let $A$ be a ring and $I$ be an ideal. I'm trying to prove that $\dim (A/I) \le \dim (A)$. My attempt to proof Suppose that $\dim (A)=n$, then there are prime ideals $\mathfrak ...
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43 views

Density of set of splitting primes

Let $K$ be a number field and let $S$ be a set of primes of $K$ containing the set of archimedian primes $S_\infty$. Suppose, $S$ has Dirichlet density $\delta(S) = 1$. Then the claim is that the ...
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Maximal and prime ideals of $\mathbb{Z} \times \mathbb{Z}$

I have to find a maximal ideal of $\mathbb{Z} \times \mathbb{Z}$ , and a prime ideal that is NOT maximal. Or, essentially, I want $I$ such that $\mathbb{Z} \times \mathbb{Z} / I$ is a field, and I ...
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Height unmixed ideal and a non-zero divisor

Let $R$ be a commutative Noetherian ring with unit and $I$ an unmixed ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal? Background: A ...
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How does one find a minimal primary decomposition?

What exactly does it mean for a primary decomposition to be "minimal" and is the a general method to obtain such decompositions? I've tried looking at some examples but they all give very little ...
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Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
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Is there an example of commutative ring with exactly three prime ideals for which this property holds?

Is there an example of commutative ring with exactly three non zero prime ideals $P_i$ which satisfies the following statement: $P_1P_2=0$ and for an ideal $I\neq 0$ such that $I\neq P_i$ we have ...
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Algebraic Geometry and Maximal ideals

I am solving the following problem but couldn't figure out a strategy to solve: Does $(x^3-17, y^2)$ generate maximal ideals in the quotient ring $R=\mathbb{C}[x,y]/I$ where $I$ is the principal ...
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46 views

Dfference between strongly prime and prime ideal

An ideal $P\subset R$ is strongly prime, if for any $x$ and $y$ in the quotient field of $R$, $xy\in P$ implies $x\in P$ or $y\in P$. What is the difference between strongly prime ideal of $R$ and a ...
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The sum of all right ideals isomorphic as modules to a simple module is an ideal

I could use some help on the following problem. Let R be a ring. (a) If $r \in R$ and $U$ is a minimal right ideal of $R$, show that either $rU=0$, or that $rU$ and $U$ are isomorphic right ...
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Isomorphism between quotient ring and its localization

Let $R$ be a domain, $P$ a prime ideal of $R$, and $k$ an positive integer. I am wondering if we have the isomorphism: $$ R/P^k\cong R_P/(PR_P)^k $$ where $R_P$ is the localization of $R$ at $P$. If ...
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Is the ideal $(x,3)$ of $\mathbb{Z}[x]$ prime or maximal?

I can't solve this question. I know that (x,3) is maximal ideal if $\mathbb{Z}[x]/ (x,3)$ is a field and (x,3) is a prime ideal if $\mathbb{Z}[x]/(x,3)$ is a domain. I know that there are isomorphism ...
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118 views

When is $\operatorname{gr}_I (M)$ finite?

When is $\operatorname{gr}_I (R)$ (I mean associated graded ring of $I$) finite? When is $\operatorname{gr}_I (M)$ finite? ($R$ is Noetherian ring and $M$ finite $R$-module.)
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Notations in Oka family definition

Definition. An ideal family $F$ in a ring $R$ with $R \in F$ is said to be an Oka family (strongly Oka family) if, for $a \in R$ and $I$, $A \lhd R$, $(I, a), (I:a) \in F \Rightarrow I \in F$ ...
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The ideal $I=(3,2+\sqrt {-5})$ is a projective module

Let $R=\mathbb Z[\sqrt{-5}]$ and $I=(3,2+\sqrt {-5})$ be the ideal generated by $3$ and $2+\sqrt{-5}$. I'm trying to prove that $I$ is a projective $R$-module. I'm using the lifting property ...
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Real analysis based on rings and ideals [duplicate]

Let $R$ be the ring of all the real valued continuous functions on the closed unit interval. Show that $ M=\{ f\in R:f(1/3)=0 \} $ is a maximal ideal
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Checking if $\langle 2 \rangle$ is a maximal ideal in $\mathbb{Z}[i]$

Is $\langle 2\rangle$ a maximal ideal in $\mathbb Z[i]$? Solution: We know that $\mathbb Z[i]$ is ED And hence PID. Consider $2\in\mathbb Z[i]$. Then $N(2)=2^2=4$ (NOTE: $N$ is norm). Since $N(2)$ is ...
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$IJ =(I\cap J)(I+J)$ holds in a PID? $I,J$ Ideals of a PID

One inequality is obvious, but the other one im not sure if holds. Any idea?
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Checking that some ideal is maximal in the multivariate polynomial ring

Let be $k$ a field and $k[x_1,x_2,...,x_n]$ its polynomial ring in $n$ variables. Let be $I$ the ideal generated by $x_1-c_1,x_2-c_2,...,x_n-c_n$, where $c_1,...,c_n$ are elements of $k$. I want to ...
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Show that a finitely generated module is trivial if it's equal to a maximal ideal times itself

Let $R$ be any commutative ring which is local. $M$ a finitely generated $R$ module, and $ I \subset R$ a maximal ideal. How to show that $ M = IM$ $ \Longrightarrow$ $M = 0$ in an elementary way? And ...
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Common divisors in a PID

Suppose that R is an integral domain and that α, β, γ ∈ R. We say that γ is a common divisor of α and β if γ|α in R and γ|β in R. Suppose that R is a PID. Suppose that α, β ∈ R. Let I = (α) and ...
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A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization ...
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Why does the only maximal of $k[[X_1,\ldots,X_n]]$ is $(X_1,\ldots,X_n)$?

I'm trying to understand in this book why the only maximal of $k[[X_1,\ldots,X_n]]$ ($k$ field) is $(X_1,\ldots,X_n)$: If I prove $rad(k[[X_1,\ldots,X_n]])\subset (X_1,\ldots,X_n)$, (where $rad$ is ...
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Quotients of Banach algebras by ideals

I am currently working through Banach Algebra Techniques in Operator Theory and am hung up on a detail on 2.32. When trying to show that the quotient of a Banach space $\mathcal{B}$ by a closed ideal ...
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Prove: The pre-image of an ideal is an ideal.

Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$.
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Proper Ideal and Units Proof

Show that an ideal of I R is proper if and only if it does not contain 1 iff and only if it does not contain any units. (1 is the identity element) I'll need to show 3 parts: (1) $\implies$ (2): ...
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Localization at a maximal ideal and quotients.

If we have a commutative ring $R$ and a maximal ideal $m$, then is $m/m^2$ isomorphic to $m_m/m^2_m$? Thx.
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Checking whether $\sqrt{2}$ is contained in the ideal $(2,\sqrt{-5}+1)$

Let $L=\mathbb{Q}(\sqrt{-5},\sqrt{2})$ and $I=(2,\sqrt{-5}+1)$ the ideal in $\mathcal{O}_L$, the ring of algebraic integers in $L$, which is generated by $2$ and $\sqrt{-5}+1$ I want to show that ...
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Question about some details of a proof of Chinese Remainder Theorem

In the proof of 3rd proposition I can prove the intersection of all ideals is the kernel of the map, but why does it imply this proposition is true?
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Principal prime ideal is generated by irreducible element

$R$ is an integral domain, $x\in R$ and $(x)=I$ is a prime ideal. Prove that $x$ is an irreducible element of $R$. So I assume $ab\in I$, with $a, b \in R$. Since $I$ is a prime ideal, either $a$ ...
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Can someone explain ideals to me?

I'm struggling with the idea of ideals (both the definitions and the common notation). I'm in a basic collegiate algebra course, just looking for a bit of help. As simply defined as possible, if you ...
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How to prove a simple proposition about local rings and maximal ideals

(The word ring shall mean a commutative ring with an identity element in this question.) Actually, there is a proof about this proposition, but I don't get it, even the first step. Proposition: ...
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Is $(X^2+Y,Y^2-2)$ maximal in $\mathbb{Q}[X,Y]$?

I'm trying to determine whether the generated ideal $(X^2+Y,Y^2-2)$ is maximal in $\mathbb{Q}[X,Y]$. I take nonzero $f\in \mathbb{Q}[X,Y]/(X^2+Y,Y^2-2)$. In this quotient, I think $Y=-X^2$, and ...
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How to write down this proof about a graded ideal in multilinear algebra?

I have a very simple question, but since this is the first time I'm dealing with graded ideals and so on it seems more difficult than it really is. Suppose $V$ is a finite dimensional vector space ...
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Are the two prime ideals containing same idempotents always the same?

If two prime ideals contain the same non trivial idempotents, what can we say about those ideals? Are they equal?
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Maximal Ideals of $\mathbb{R}^{\infty}$

In the ring $\mathbb{R}^{\infty}$ (with the standard operations of component-wise addition and multiplication), what are the maximal ideals? It was quite simple to determine that the ideals with a ...
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No units in quotient ring equivalent to no units in original ring?

Definition: Let $R$ be a ring with $1$. $r\in R$ is a unit if and only if $r \neq 1$ and there exists $s\in R, s \neq 1$ such that $rs=1=sr$. Let $R$ be a ring with $1$ and let $I$ be a proper ideal ...
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What does $J_1\cap J_2=\emptyset$ mean algebraically for two varieties in $\Bbb{C}^n$?

Let $J_1, J_2$ be two varieties in ${\Bbb C}^n$. Then $$ J_i=V(I_i)\quad i=1,2. $$ for some $I_i\subset\Bbb{C}[x_1,\cdots, x_n]$ and $$ J_1\cap J_2=V(I_1\cup I_2) $$ and $$ J_1\cup J_2=V(I_1I_2). $$ ...