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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Saturation of homogeneous ideal

Let $I \subset S=k[x_0,...,x_n]$ be a homogenous ideal. The saturation of $I$, $\bar{I}$ is defined to be $\{s \in S: \exists m \; s.t. \; \forall i \; x_i^m s \in I\}$ Is it true that $\bar{I}=\{s ...
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Given two polynomials $f$ and $g \in \mathbb{Q}[X]$, prove that $(f) + (g) = (h)$ and $(f)\bigcap(g) = (k)$

Given two polynomials $f(X) = 3X^2 + 7X - 6$ and $g(X) = 2X^2 + 5X - 3 \in \mathbb{Q}[X]$, prove that there exist $(h)$ and $(k) \in \mathbb{Q}[X]$ such that $(f) + (g) = (h)$ and $(f)\bigcap(g) = ...
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Showing the ideal $\left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ is radical for all $a\neq 0$

Let $I_a = \left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ be an ideal of $k[x,y,z]$, where $a \neq 0$. Show that $I_a$ is radical. What is the geometric meaning of the elements in ...
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Showing Quotient ring is a field using maximal Ideal

Question: Show that $R\left [ x \right ]/\left \langle x^{2}+1 \right \rangle$ is a field. Recall: Theorem: Let R be a commutative ring R with unity. Let I be a proper Ideal of a ring ...
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Verify size of factor ring

Let the ring $R=\left \{ \begin{bmatrix} a_{1} &a_{2} \\ a_{3}& a_{4} \end{bmatrix} \mid a_{i} \in \mathbb{Z} \right \}$ and let I be the subset of R consisting of matrices with even ...
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$B/I$ and $B/J$ flat $A$-algebras; does $I=J$ hold?

Let $A\to B$ be a ring homomorphism. Consider $I$ and $J$ ideals of $B$ such that $B/I$ and $B/J$ are flat $A$-algebras. We know furthermore that there exists a non zero-divisor $t\in A$ such that ...
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Any ideal of a field $F$ is $0$ or $F$ itself

Prove that the only ideals of a field are $\left\{ 0 \right\}$ and the field itself. Let $F$ be a field and $I$ be an Ideal of $F$. Let $0 \ne x \in I$. Since $I$ is an Ideal of $F$, it is true ...
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24 views

Singular ideals and rings

In Lam's book, Corollary (7.4)(2) says that for a nonzero ring $R$ we have $Z(R_R)≠ R$, where $Z(R_R) $ stands for the singular ideal of $R$.. But, some nonzero commutative rings are "singular" in the ...
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52 views

Is there a way to characterize the prime ideals in $\mathbb{R}[x_1,x_2, \dots , x_n]$?

I'm studying algebras which can be formed by the quotient of principal ideals in $\mathbb{R}[x_1, \dots , x_n]$, and thus would like to be able to determine which of said principal ideals are maximal, ...
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A modularity condition

Let $R\subseteq S$ be rings with unity and $X$ ,$Y$ be subsets of $S$ with $X$ an ideal. If $S=X+Y$ what conditions should be held to infer the equality $R=(X∩R)+(Y∩R)$? I think that if we ...
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Prove that $M_p$ is a ideal of $\mathbb Z/(p)[x]$ and $\mathbb Z[x]/M$ is isomorphic to $\mathbb Z/(p)[x]/Mp$.

Let $M_p$ = $\gamma (M)$, the image of $M$ ($M$ is a maximal ideal of $\mathbb Z [x]$) in $(\mathbb Z/(p))[x]$, where $\gamma$: Z[x] --> Zp[x] is the morphism such that $\gamma (\sum_i a_ix^i)=\sum_i ...
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20 views

Showing an Ideal is the ring

If A is an Ideal of a ring R and the unity 1 belongs to A, prove that A=R. It is a sufficient condition to show that $A\subseteq R$ and $R\subseteq A$. Indeed, it is trivial to see that ...
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Intersection of any set of ideals is an ideal

Prove that the intersection of any set of Ideals of a ring is an Ideal. I'm looking for hints. Let A, B both be Ideals of a ring R. Suppose $I \equiv A\cap B$. Since A and B are both Ideals of ...
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26 views

Proof Verification of Result Involving Maximal Ideals

In further investigation of a question I asked earlier, I came across the following result, the proof of which I hope can be looked over here. I personally find it kind of interesting and I hope ...
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110 views

Problem about Gröbner basis.

I'd really appreciate if someone could help me. The problem is the following: Let $\psi_1,...,\psi_m \in k[x_1,\dots,x_n]$ and consider the $k$-algebra homomorphism: ...
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Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$, prove that $\mathrm{Spec}(A_f)=\mathfrak{V}((f))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in ...
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104 views

Primary descomposition of ideals

I'd appreciate if someone could help me a bit with this problem. Considering $\mathfrak{p}=(x,y), \mathfrak{q}=(x,z)$ and $\mathfrak{m}=(x,y,z)$ ideals in $k[x,y,z], k$ field. Is ...
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Correspondence between nilpotents and between idempotents

It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with ...
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Maximal (inclusion) ideal proof

Let $X$ be a set and $I$ its ideal. $I\neq \emptyset$ is ideal if $I\subseteq \mathcal{P}(X)$, so that for all $A,B\in \mathcal{P}(X)$ following holds $$(A\subseteq B \text{ and } B\in I)\Rightarrow ...
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49 views

localized at associated prime of an ideal [duplicate]

The problem is as follows: Let $I\subseteq J$ be ideals in a Noetherian ring. Show that if $I_{p}=J_{p}$ for every associated prime $p$ of $I$,then $I=J$. It seems reasonable to consider ...
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Integrally Closed domain and Principal Ideal

Let $R$ be an integrally closed local domain. Suppose there is a $y\in I^n$ such that $yI^n=I^{2n}$ for some $n$. I would like to prove that $I^n=(y)$. Source: The above question comes from the ...
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Prove that in a Euclidean domain nonzero prime ideals are maximal. [closed]

It is from masters qualifying exam. I am an undergraduate student. I want to wonder this proof. Can you prove to explain clearly?
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130 views

How can we find the prime ideals of $\mathbb{Z}_{12}$? [closed]

I have found that the maximal ideals of the ring $\mathbb{Z}_{12}$ are $(2)$ and $(3)$. Is this correct? How can we find the prime ideals of $\mathbb{Z}_{12}$ ?
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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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86 views

$\mathbb{Z}[x]$ doesn't have principal maximal ideals [closed]

Prove that $\mathbb{Z}[x]$ doesn't have principal maximal ideals. Please, I need help with this problem. Thanks!
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1answer
44 views

Existence of homogeneous non-unit non-zero divisor in a particular graded ring.

Let $R$ be a finitely generated $k$-algebra of dimension greater than $1$, let $Q$ be any maximal ideal of $R$. It is claimed by my lecturer that one can find a homogeneous, non-unit, non-zero divisor ...
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50 views

Easy explanation on primary decomposition of ideals. [duplicate]

The primary decomposition of an ideal $(x^2, xy)$ is $$(x^2, xy) = (x) \cap (x, y)^2$$ which can be found on these notes. Could someone explain to me how this can be done? Edited: My question ...
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Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field. I am not looking for a solution here, rather a hint or two. Is there a general strategy for ...
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Ideal of a Lie Algebra

I was given this, I think unusual, definition of ideal of a Lie algebra: a subset $I$ of a Lie algebra $L$ is called an ideal if $[I,L]\subseteq I$. I was told from this follows that $I$ is a ...
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Prove the Radical of an Ideal is an Ideal

I am given that $R$ is a commutative ring, $A$ is an ideal of $R$, and $N(A)=\{x\in R\,|\,x^n\in A$ for some $n\}$. I am studying with a group for our comprehensive exam and this problem has us stuck ...
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Simple Question about Valuations and Krull Rings

I have what is a very simple question about essential valuations for Krull rings. Before getting to the question, I'll give a sketch of the situation. Any help would be much appreciated. Suppose that ...
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42 views

Maximal ideals in a ring of sets

A ring $R$ is called Boolean if $x^2 = x$ for all $x \in R$. It follows that Boolean rings have characteristic $2$ and are commutative. Let $S$ be a non-empty set, then $P(S)$ with $A + B = (A - B) ...
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$k[x,y]/(xy-1)$ isomorphic to $k[x,\frac{1}{x}]$ [duplicate]

I mean clearly one uses the isomorphism $\phi$ that sends to $x$ to $x$ and $y$ to $\frac{1}{x}$. And also clearly is $(xy-1)\subseteq\ker(\phi)$. I just struggle to prove the other inclusion. Can ...
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Show that $(X_{k_1}^{a_1},…,X_{k_s}^{a_s})$ is $(X_{k_1},…,X_{k_s})$-primary [duplicate]

Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary, where $I$ is the ideal generated by the monomials $X_{k_1}^{a_1},...,X_{k_s}^{a_s}$ . I noticed that ...
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Example of an ideal which is not principal in the ring $\mathbb{Z} [x]$ [duplicate]

Give an example of an ideal in the ring $\mathbb{Z} [x]$ is not principal. What kind of example would be the easiest?
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Extension of intersection of ideals

Let $f:A \rightarrow B$ be ring homomorphism and $\mathfrak{a}_1,\mathfrak{a_2}$ be ideals of $A$. Let $\mathfrak{a}^e$ denote the extension of an ideal $\mathfrak{a}$ of $A$ in $B$. An exercise shows ...
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1answer
60 views

Flatness and intersection of ideals [closed]

This is Exercise 1.2.6(a) in Liu, Algebraic Geometry and Arithmetic Curves Let $B$ be a flat $A$-algebra. Show that for any finite family $\{I_\lambda\}_{\lambda\in \Lambda}$ of ideals of $A$, ...
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64 views

How to prove that two principal ideals are equal [closed]

Background info Provided that the general form of a polynomial is $(a_0X^{0}+...+a_nX^{n})$ where $X$ is an element of the field provided. an ideal generated by an element is the set $(a*r s.t. ...
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Relationship between modules and maximal ideals of a commutative ring

Let $A$ be an integral domain, $M$ an $A$-module, and $m\in M$. Now for all maximal ideals $\mathfrak{m}$ there exists an $n\notin \mathfrak{m}$ such that $nm=0$. Why does this mean that $m=0$?
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Maximal ideal of subalgebra over a field [duplicate]

Let $A$ a finite $k$-algebra (with $k$ a field) and $B$ a subalgebra of $A$. Prove that if $\mathfrak{m}$ is a maximal ideal of $A$ then $\mathfrak{m}\cap B$ is a maximal ideal of $B$. It is easy ...
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A nilpotent Jacobson radical?

If each ideal of a commutative ring $R$ could be written as a sum of a nilpotent ideal $N$ and an idempotent ideal $I$, is the Jacobson radical $J(R)$ of $R$ necessarily nilpotent (or T-nilpotent)? ...
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Annihilator - Product of cyclic groups

Let $M$ be the abelian group, i.e., a $\mathbb{Z}$-module, $M=\mathbb{Z}_{24}\times\mathbb{Z}_{15}\times\mathbb{Z}_{50}$. I want to find the annihilator $\text{Ann}(M)$ in $\mathbb{Z}$. $$$$ ...
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Find a polynomial $g(x) \in \Bbb Q [x]$ such that $I = g(x) $

Find a polynomial $g(x) \in \Bbb Q [x]$ such that ideal $I = (g(x)) $, where $I = \{f(x) \in \mathbb Q[x] : f(\sqrt2) = 0\}$ $ I = \{f(x) \in \mathbb Q[x] : f(1-i) = f(1+i) = 0 \}$ For 1, I think ...
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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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Can a Nonzero Element in $\mathbb Z[\omega]$ be Divisible by Arbitrarily Large Powers of $1-\omega$.

Question. Let $p$ be a prime and $\omega$ be a primitive $p$-th root of unity. Let $a$ be a nonzero element of $\mathbb Z[\omega]$. Can it happen that for each $n\in \mathbb N$, $(1-\omega)^n$ ...
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1answer
29 views

A uniqueness theorem for primary decomposition

"Let $R$ be an arbitrary ring and $\mathfrak a$ an ideal of $R$ admitting an irredundant primary representation $\mathfrak a=\bigcap_{i}\mathfrak q_{i}$ and let $\mathfrak p_i=\sqrt{\mathfrak q_i}$. ...
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Bases of free module of rank $2$

The following problem might be as simple as it looks. Let $R$ be an integral domain, $M$ a free $R$-module of rank $2$, and $\{m_{1},m_{2}\}$ a basis for $M$. Suppose that $$ x = ...
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1answer
62 views

How many ideals in a ring R turned into Z/nZ

Say I have a ring R, is there any general way to find out how many ideals it has? I know that if it's a field then there are only 2 ideals, namely (0) and (1), however what if the ring is not a field, ...
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1answer
53 views

Filling in Proof: Well-definedness of depth(I,M).

From Eisenbud's Commutative Algebra with A View Toward Algebraic Geometry (Theorem 17.4): Let $M$ be a finitely generated $R$-module, where $R$ is Noetherian. If $$r= \min \{i : H^i(M\otimes ...
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1answer
39 views

Ideal quotient is principal

Let $R$ be a commutative ring with $1$ and $I$ an ideal. Also let $B$ be a principal ideal, and $A=\{a\in R\;|\; aB\subseteq I\}$. I want to show that if $A$ is also principal then $I$ is principal. ...