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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Minimal prime ideals consist of zerodivisors [duplicate]

I don't find the proof for this little demonstration ... Let $P$ be a minimal prime ideal of $A$. Show that $P$ is contained in the set of zero divisors of $A$.
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Ideal of a Vanishing set $I(V(F[X,Y]))$ and how to repeat the computation.

The video I am getting this from is found here: https://www.youtube.com/watch?v=spHxUPvrkXw, it is around 5 minutes in. The first part of the question is: for $F[X,Y] = Y^2 - X^3 = 0$ find ...
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71 views

Invertible elements and maximal ideals of a localization

Let $n\in\mathbb Z$ and let $A$ be the set of integers co-prime to $n$. Denote $A^{-1}\mathbb Z$ by $\mathbb Z_{(n)}$. 1) Find the invertible elements of $\mathbb Z_{(6)}$ My attempt: let $m$ be ...
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Question about comaximal ideal proof

Let $A$ be a ring and $M\subseteq A$ a maximal ideal. Show that if $I\subseteq A$ such that $I\not\subseteq M$, then $M$ and $I$ are comaximal($M+I=A$). I cannot find the proof for this statement.
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Show that these rings of Gaussian integers are ideals in $\mathbb{Z}[i]$?

Consider the ring of Gaussian integers: $\mathbb{Z}[i]$ = {a + bi | a, b ∈ Z} ⊂ $\mathbb{Q}[i]$ with $i^2$ = −1. Let I = $(2+3i)$ and J =$(2−3i)$. Show that I and J are ideals of $\mathbb{Z}[i]$.
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Example of irreducible ideal which is not strongly irreducible

I have read a paper with title Ideal Theory in Commutative Semirings by Reza Ebrahimi Atani and Shahabaddin Ebrahimi Atani. In this paper we have the following definitions: An ideal I is irreducible ...
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71 views

Maximal ideals in $\mathbb{Z}[x]$

I am trying to solve the following problem from Artin: Every maximal ideal $\mathbb{Z}[x]$is of the form $(p,f)$ where p is a prime integer and $f$ is a primitive polynomial that is irreducible modulo ...
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Class Group of Ring of Integers of $\mathbb{Q}[\sqrt{-57}]$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need ...
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Normal ring and unmixed ideals

Let $R$ be a commutative Gorenstein local ring , $I$ an ideal of $R$ . If $R/I$ is normal ring , then for any $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?
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If $A/\mathfrak a$ is flat over $A$ then $V(\mathfrak a)$ is open. Why?

I am trying to understand the following statement. Let $A$ be a noetherian commutative ring and $\mathfrak a\subset A$ is an ideal. Suppose that the ring $A/\mathfrak a$ is flat over $A$, then ...
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64 views

Ideals in direct product of rings

I am trying to solve that problem: Let $ R_1,...,R_n$ rings with identity, every ideal of $R=\prod_{i=1}^n R_i$ is in form $\prod_{i=1}^n I_i$ where $ I_i$ideal of $R_i$. The first part is clearly if ...
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68 views

Minimal primes and zero divisors

Let $R$ be a commutative local ring, $M$ a finitely generated $R$-module, and $x \in M$. Is it true that if for any $p \in$ $\operatorname{Min}(R)$ there exists $a_{p}\notin{p}$ such that $a_{p}x=0$, ...
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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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56 views

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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36 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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54 views

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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80 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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60 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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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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1answer
60 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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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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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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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
145 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
43 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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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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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
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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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82 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
60 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
49 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}) = ...