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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Question on radical ideals

i need help with showing that I = <xy, xz, yz> is a radical ideal. Thanks
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Direct product of algebras over a field

Let $ B_1,B_2,...,B_n$ k-algegras, $ B=\prod_{i=1}^{n}B_i $ the direct product of those (k is a field) , and $ J_i$ an ideal of its k-algebra. i must to prove that: The direct product $ ...
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Let $I= (X_1,X_2) \cap (X_3,X_4)$. Is $ara(I)≥3$? Is $ara(I)≥4$?

This question is related to Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$ Let $R=k[X_1,X_2,X_3,X_4]$ and $I= (X_1,X_2) \cap (X_3,X_4)$. I know that ...
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Give an example of an ideal in $\mathbb{Z}\times\mathbb{Z}$ which is maximal.

My answer right now is just $(0,1)$ and $(1,0)$ resulting in $\mathbb{Z}\times\mathbb{Z}$ as $(1,1)$. But this is the entire ring... Help?
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If $I\leq K[X_0,\dots,X_n]$ for $K$ a field is an ideal whose radical is homogeneous, is it always the case that $I$ is homogeneous?

If $I\leq K[X_0,\dots,X_n]$ (for $K$ a field, let's say algebraically closed) is an ideal whose radical is homogeneous, is it always the case that $I$ is homogeneous? I'm trying to understand ...
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Associated primes and their heights

Let $(S,m)$ be a commutative Gorenstein local ring, $I$ an ideal of $S$ such that $\operatorname{ht} I=t$, and $R=S/I$. Let $a \in m$ be an $R$-regular element such that for any prime ideal ...
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43 views

Prime ideals in formal power series

Let $A$ be a commutative ring with unit. If $\mathfrak{p} \subset A $ is a prime ideal, then $\mathfrak{p}$ is the contraction of a prime ideal of $A[[x]]$, the ring of formal power series. Why is ...
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Ring theory question: $I=\langle x,2 \rangle$ prime/maximal ideal in $\mathbb Z[x]$?

In $\mathbb{Z}[x]$ , let $I = \lbrace f(x) \in \mathbb{Z}[x] : f (0) \text{ is an even integer} \rbrace.$ Is $I=\langle x,2 \rangle$ a prime ideal of $\mathbb{Z}[x]$? Is $I=\langle x,2 \rangle$ a ...
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115 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
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Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
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Maximal element of $(I : x)$, where $x$ is in $A - I$, is prime belonging to $I$

Given that $I$ is decomposable, I am supposed to prove that any maximal element $P$ of the set {$(I : x) | x \in A - I$} must belong to $I$, i.e., $P$ is prime and for every reduced primary ...
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Properties induced by surjective ring homomorphism between two integral domains

Let $f: R \rightarrow S$ be surjective ring homomorphism between two integral domains. (a) If $R$ satisfies Ascending Chain Condition on Principal Ideals, must $S$ also satisfies Ascending Chain ...
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In $\mathbb{Z}[t]$, $Q = (4, t)$ is not a power of $M = (2, t)$

The problem of showing that Q, as above, is not a power of M, as above, rises as part of a larger problem. I'm confident about my response to the other parts, but the best justification I can come up ...
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31 views

Finitely generated ideal in boolean ring [duplicate]

A boolean ring is a commutative ring where $x^{2} = x$ for every $x$. Why in such a ring a finitely generated ideal is principal ?
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The ideal $I=\{a_{0}+a_{1}x+\cdots+a_{k}x^{k} \in F[x]\mid a_{0}+a_{1}+\cdots+a_{k}=0\}$ equals $\langle x-1\rangle$?

If $I=\{a_{0}+a_{1}x+\cdots+a_{k}x^{k} \in F[x]\mid a_{0}+a_{1}+\cdots+a_{k}=0\}$ is an ideal in $F[x]$, is it equivalent to $I=\langle x-1\rangle$? Think it is because $1$ is obviously a root of ...
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Idempotent in a local ring

Is it true that a local ring, i.e. a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
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Height unmixed ideals

Let $R=k[x_1,\ldots,x_n]$ be a standard graded polynomial over field $k$ and $I$ an unmixed homogeneous ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an ...
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Primary ideals in Noetherian rings

For an $R$-module $M$ I have the following definition for a submodule $N\subset M$ to be $\mathfrak{p}$-primary: this is the case when $\text{Ass}(M/N) = \{\mathfrak{p}\}$. This equals saying that ...
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1answer
32 views

Maximal (among non-principal ideals) Ideal Must be Prime

If I is an ideal which is maximal among the ones that are not principal, then I is prime. This would mean that for all $f \in R$, $(f) \subset I$. Could I then use column ideals? I was thinking ...
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30 views

Maximal Ideal Must be Prime

I am trying to prove that an ideal that is maximal with respect to not being finitely generated must be prime. What does it mean to be an ideal that is maximal with respect to not being finitely ...
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56 views

Zero dimensional ideals and their primary decomposition

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$, and $I$ a zero dimensional ideal with a primary decomposition $I=\cap Q_i$. Why is $\sum \dim_k S/Q_i = \dim_k S/I$?
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Find the projective closure of the ideal $I=\langle y-x^2,z-x^3\rangle$

When I looked at this example, my first instinct was to homogenize only the generators of $I=\langle f_1 := y-x^2,f_2:=z-x^3\rangle$ in a new variable $w$. But then, I realized that I would miss some ...
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Annihilator of $a'$ and $b'$ in the ring $\mathbb{Z}/(a'b')$ ?

I want to find the annihilator of $a'$ and $b'$ of the quotient ring $R=\mathbb{Z}/(a'b')$ where $a',\,b'>1$. So if I go by the definition, $ann(a')=\{r\in\mathbb{R}\mid ...
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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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50 views

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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51 views

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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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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1answer
36 views

$\dim (A/I) \le \dim (A)$

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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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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1answer
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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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59 views

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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1answer
55 views

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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1answer
42 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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50 views

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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1answer
88 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? ($M$ is an $R$-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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1answer
67 views

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 ...