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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Some equivalences for Ideals of the ring of real valued functions

Let $I_{N}=\{f\in\Omega(\Bbb R,\Bbb R)|\forall x \in N\subset \Bbb R: f(x)=0\}$, where $\Omega(\Bbb R,\Bbb R)=\{f:\Bbb R \to \Bbb R|f\space is \space a\space function\}$, then the following statements ...
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Why the ideal norm is multiplicative

Let $I\subseteq B$ be an ideal, we define the ideal norm of $I$ as the ideal in $A$ generated by the elements $N_{E/K}(\alpha)$ where $\alpha \in I.$ We denote it by $N_{E/K}(I).$ If $\mathfrak{p}$ ...
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Question about radical ideal

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1,...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], \deg(f) \...
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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 $(f)\bigcap(g) = (k)$...
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42 views

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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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 R. ...
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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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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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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 $\sqrt{I_0}\...
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$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 $(B/...
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60 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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21 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 $A\subseteq ...
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27 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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30 views

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 a ...
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28 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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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 A\...
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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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93 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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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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46 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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35 views

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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1answer
63 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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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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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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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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Relationship between modules and maximal ideals of a commutative ring [closed]

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 to ...
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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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1answer
56 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 $J/I\...
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$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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1answer
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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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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 = s_{1}m_{...
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1answer
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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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A conceptual question in ring theory?

What is the main(conceptual) difference between an ideal of a ring and a submodule over a ring?
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To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F[x,y]$ by showing that $F[x,y]/\langle x-a , y-b\rangle$ is a field [duplicate]

Is there any way to show that for $a,b \in F$ , the ideal $\langle x-a , y-b\rangle$ is maximal in $ F[x,y]$ , by showing that the quotient $F[x,y]/\langle x-a , y-b\rangle$ is a field ? Is the ...
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How else can I tell I can do this with $5$ but not $2$ or $3$ in $\textbf{Z}[\sqrt{30}]$?

In $\textbf{Z}[\sqrt{30}]$, the number $5$ splits, since, for example, $N(5 + \sqrt{30}) = -5$. But the ideal $\langle 5 \rangle$ is a ramifying ideal, since it is equal to $\langle 5, \sqrt{30} \...
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Buchberger algorithm and ideals

I'm working on Groebner bases using the book Ideals, Varieties and Algorithms. I'm interested in this problem : Let $\mathbb{Q}[x,y,z]$ with the graded lexicographic order with $x>y>z$. For ...
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Are the following two ideals equal? How to prove it, or show they are not?

$I= \langle x-y^2, x-y^3, x-y^4,... \rangle, $ and $J=\langle x-y^2, x-y^3\rangle$. Obviously $J \subset I$, but what about the reverse inclusion?
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Ideal of a product ring?

I am trying to prove whether or not the ideal generated by $\langle (2,2)\rangle$ is a prime ideal of $\mathbb Z_4\times \mathbb Z_4$? My issue is I'm not sure how to do the coordinate ...
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Example of associated ideal in primary decomposition

Let $I$ be a decomposable ideal of a commutative ring $R$ with minimal primary decomposition $I=\bigcap_{i=1}^n\mathfrak q_i$. The first uniqueness theorem shows that $\{\sqrt {\mathfrak q_i}:1\le i\...
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63 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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32 views

Regarding taking powers of prime ideals in a ring

My question is simple to ask: given some prime ideal $P$ in a ring $R$, we can talk about $P^2, P^3$ etc. but can we discuss $P^0$? Is there a convention that says $P^0 = R$, or is there something ...
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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 K(x_1,...
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Prime and maximal ideal

If I have to show that ideal A is not maximal, is it enough to show that A is not prime because it is usually easier? Every maximal ideal is prime so if we have ideal that is not prime, it can not be ...