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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Spec($A$) is connected if $A$ is local

Another exercise from Balwant-Singh: Show that if $A$ is local then Spec($A$) is connected in the Zariski topology. Any hint ?
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Idempotent/Spec

I'm studying Basic Commutative Algebra by Balwant-Singh; I'm stuck on this exercise: $A$ is a commutative ring; show this $3$ conditions are equivalent: 1) $A$ contains a non-trivial idempotent 2) ...
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Is it true that an ideal is primary iff its radical is prime?

Is it true that an ideal $I$ in a commutative ring is primary iff $Rad(I)$ is prime? If not, what are some nice counterexamples?
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41 views

When an Intersection of Prime Ideals is a Prime Ideal

Let $R$ be an arbitrary ring, $\{P_1,....,P_n\}$ be a set of prime ideals. Verify that $P_1 \cap ... \cap P_n$ is prime if and only if there exists $1 \leq i \leq n$ such that $P_i$ is contained in ...
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Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
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Complexifications of degree 3 subschemes in $\mathbb A^2_{\mathbb R}$

I am trying unsuccessfully to solve exercise II-20 (page 65) from the book "The geometry of schemes" by Eisenbud and Harris. In this exercise it is stated that there are two non-isomorphic subschemes ...
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Prime ideal in a ring with some property is maximal

Let $R$ be a ring where for every element $a \in R$, there exists a positve integer $n_a \gt2$ such that $a^{n_a}=a$. Prove that every prime ideal in $R$ is maximal. I think that I would want to ...
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1answer
83 views

When does coprimality carry over to the base ring in an extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
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88 views

A monomial ideal, $I =\langle xy, xz, yz\rangle$, is radical

I need help in showing that $I =\langle xy, xz, yz\rangle$ is a radical ideal. Thanks
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65 views

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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78 views

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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1answer
95 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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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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1answer
88 views

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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0answers
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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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1answer
34 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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Idempotents 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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1answer
131 views

Height unmixed homogeneous ideal and a non-zero divisor

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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1answer
151 views

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 $\text{Ass}(M/N) = \{\mathfrak{p}\}$, that is, $M/N$ is coprimary ...
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1answer
45 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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1answer
41 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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70 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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1answer
89 views

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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1answer
101 views

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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1answer
62 views

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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1answer
50 views

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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1answer
84 views

$\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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0answers
68 views

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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1answer
61 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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1answer
94 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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1answer
63 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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1answer
47 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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1answer
45 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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1answer
70 views

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 ...
5
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1answer
95 views

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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2answers
39 views

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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1answer
37 views

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 ...
4
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1answer
158 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 ...