Questions tagged [ideals]

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, ideals in Lie algebras.

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Short Question: $(p)$ for a prime is not a maximal ideal in $\mathbb{Z}[X]$

Given a prime number $p\in\mathbb{Z}$ I want to show that $(p)$ is not a maximal ideal in the ring of polynomials $\mathbb{Z}[X]$. I know how the maximal ideals in $\mathbb{Z}[X]$ look like, I want ...
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How do you prove that 2 nontrivial ideals are the only nontrivial ideals in a ring?

What do you need to look for in 9rder to prove that 2 nontrivial ideals are the only nontrivial ideals in the ring? I can prove that if my ideals are I and J that I+J=R, where R is the ring, but I don'...
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About countable boolean algebras

my question is concerning the article from book "countable boolean algebras and decidability", Goncharov. here we sat homomorphism from A to B image 1 here we define composition of ideals (I×J), and ...
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Lemma for Chinese Remainder Theorem

I want to prove the following Lemma: Let $R$ be a commutative ring with $1\neq 0$ and $I,J_{1}, J_{2},..., J_{n}$ be the ideals of $R$ such that $I+J_{i}=R$ $\forall 1\leq i\leq n$. Then, $I$, $J_{1}...
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Understanding the Definition of Ideals Generated by Polynomials

I would like to confirm whether my interpretation of the definition of ideals generated by polynomials is correct, please. In Ideals, Varieties, and Algorithms, Cox et al. define this as such (...
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Find an ideal $I$ in $A$ so that $A/I$ is a finite field of $25$ elements.

Let $A = \frac {\Bbb Z[X]} {\left ( X^4+X^2+1 \right )}.$ Find an ideal $I$ in $A$ such that $A/I$ is a finite field of $25$ elements. I have seen that the polynomial $X^4+X^2+1$ is reducible in $\...
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Inverse of a finitely generated ideal in UFD

Let $R$ be a UFD and $K$ be its field of fractions. Let $A$ be an ideal of $R$. Define for this $A$, the $R$-submodule $A^{-1}$ of $K$ given by $$A^{-1}=\{ \alpha\in K \,\,:\,\, \alpha A\subseteq R\}....
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For ideals of $R$, $I_1, I_2$, Is it true that $(I_1 \cap I_2)M \cong I_1 M \cap I_2 M$ for any $R$-module $M$?

It seems true if $M$ is finitely generated module over PID $R$ because we can take a linearly independent finite generator and so we can use $$ \sum_{i=1}^n c_i m_i = 0 \Leftrightarrow \forall i \in \{...
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Decomposition of semi-simple Lie algebra into simple lie algebra (or ideal?)

A semi-simple Lie algebra $L$ can by definition be decomposed into simple Lie algebras : $L=L_1\oplus \ldots \oplus L_n $. Are these $L_i$ necessarily ideals of $L$?
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Prove that $\left( I_1I_2 \right)^e=I_1^e I_2^e$

I have a problem when proving a basic exercise about extension in commutative ring: Let $f:A \rightarrow B$ be a ring homomorphism, $I_1$ and $I_2$ are ideals of $A$. Prove that $\left( I_1I_2 \...
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show that a kernel is principal, here the application : $ \text{ev}_{x = t^2, y = t^3} $

$\mathbb C [x,y] \to \mathbb C [t]$ with the evaluation : $$ \text{ev}_{x = t^2, y = t^3} $$ How can you show that the kernel is principal in $\mathbb C[x,y] $? I think the kernel is the polynomial ...
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Converse of : If $M$ is Noetherian then $M_m$ is Noetherian for every $m\in\operatorname{Max}(R)$.

Let $R$ be a Noetherian ring and $M$ be an $R$-module. We know that if $M$ is Noetherian then $M_m$ is a Noetherian $R_m$-module for every $m\in\operatorname{Max}(R) $. Now, I want to know if the ...
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$\mathfrak{q}$ is primary iff: given $a,b \in A$, $ab \in \mathfrak{q}$ and $a \not\in \mathfrak{q}$, then $b^n \in \mathfrak{q}$ for some $n \geq 1$

I am studying from Lang's Algebra, and in Chapter X Noetherian Rings and Modules, $\S$3 Primary Decomposition, he makes the following definitions on page 421, third edition (assume that $A$ is a ...
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why is a ring ideal not called a filter

A ring ideal can be characterized by the two rules: $$(a\in I) \wedge (a ~ \textrm{divides} ~ b) \implies b \in I$$ $$ a,b \in I \implies \textrm{gcd}(a,b) \in I$$ (the usual definition states $a,b \...
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For Pairwise Comaximal ideals $I_1, …, I_n$, $I_1 \cap…\cap I_n \subset I_1I_2 \cdots I_n$

In a set of Pairwise Comaximal $I_1, ..., I_n$ ideals of a commutative ring, $I_1 \cap...\cap I_n \subset I_1I_2 \cdots I_n$. I get how to do it for the case when $n=2$: choose $a \in I, b \in J,s.t. ...
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$(0)$ and $p^n\mathbb{Z}$ (where $p$ prime, $n$ positive integer) are the only primary ideals in $\mathbb{Z}$

I am trying to show that $(0)$ and $p^n \mathbb{Z}$ are the precisely primary ideals in $\mathbb{Z}.$ Clearly $(0)$ is a prime ideal hence primary and radical of $p^n \mathbb{Z}$ being the maximal ...
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In $\mathbb Z[x]$, is $(2,x)=(2)+(x)$?

The text says that $(2,x)=(2)+(x)$, because $1 \in \mathbb Z$. I do not see why this leads to the decomposition. Can someone point me in the right direction?
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Find an ideal $I$ of $\mathbb Z/2\mathbb Z[x]$, such that $\mathbb Z/2\mathbb Z[x]/I$ is a field of 8 elements.

My attempt: A field can be constructed by taking the quotient by a maximal ideal. $\mathbb Z/2\mathbb Z[x]$ is a PID, so a maximal ideal can be made by taking an ideal generated by an ...
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When do (multivariate) polynomial rings fail to be Prüfer rings?

In what follows, ring is defined to be a commutative ring with unit ($1$). Definition: Perhaps over-generalizing from (12) in this Math.SE answer, call any ring $R$ a Prüfer ring if, for all non-zero ...
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If all submodules of a free $R$-module is free, then $R$ is a PID. [duplicate]

Let $R$ be a commutative ring with identity. Prove that if all submodules of a free $R$-module is also free, then $R$ is a PID. I have managed to prove $R$ is an integral domain. For the PID part, by ...
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Finding monic generator for ideals generated by polynomials of $\mathbb{Q}[X]$

Let $R = \mathbb{Q}[X]$ be the ring of polynomials with rational coefficients and let: $$I = \langle X^2 + 1 \rangle = \{(X^2 + 1)f(X) \ \vert \ f \in\mathbb{Q}[X] \} $$ $$J = \langle 2X + 1,...
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Ideal generated by two relatively prime polynomials

Let $f(x,y),g(x,y)\in\mathbb R[x,y]$ such that $\gcd(f(x,y),g(x,y))=1$. Let also $I\triangleleft\mathbb R[x,y]$ be the ideal generated by $f(x,y), g(x,y)$. $I=\langle f(x,y),g(x,y)\rangle$ What ...
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Nonzero prime ideal containing no other nonzero prime ideal

Let $R$ be a UFD and $P$ be a nonzero prime ideal of $R$. Suppose that $P$ does not contain any nonzero prime ideal other than $P$. What can you say about $P$? I think $P$ should be a principal ideal....
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Quadratic field ideal find $\mathbb{Z}$-basis given a $\mathcal{O}_K$-basis

Suppose we are working in an imaginary quadratic number field $\mathbb{Q}(\sqrt{d})$ (so $d$ is a fundamental discriminant with $d < 0$). Now in the ring of integers $\mathcal{O}_K$ suppose we ...
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The ideal generated by a maximal orthogonal system in a Banach lattice

I have a pretty specific question about H.H. Schaefer's "Banach lattices and positive operators" book. In chapter 3, part 6 (page 169), it is said that the ideal generated by a set S (which is a ...
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Find a finite Gröbner basis for ideal $I \subseteq \mathbb{R}[x, y, z]$

Find a finite Gröbner basis in lexicographic ordering $x \prec y \prec z$ for ideal $I \subseteq \mathbb{R}[x, y, z]$, where $$ I = \{ f \in \mathbb{R}[x, y, z] \space | \space f(a, -a, 2) = 0 \...
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Show that if $I$ is an ideal contained in $J$ then $(R/I)\setminus (R/I)^{\times}$ is an ideal of $R/I$

Question: Let $R\neq 0$ be a ring and $J= R\setminus R^{\times}$ an ideal of $R$. Show that if $I\subset J$ is an ideal of $R$ then $(R/I)\setminus (R/I)^{\times}$ is an ideal of $R/I$ Attempt: I ...
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Singularity of a curve in higher dimensions

Although the problem I am working on is regarding a curve in 6 dimensional affine space, for simplicity, let's consider the Viviani's curve in $\mathbb{C}^3$: $$\mathcal{I}=\langle {x}^{2}+{y}^{2}+{z}^...
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Why is the Killing form of $\mathfrak{g}$ restricted to a subalgebra $\mathfrak{a} \subset \mathfrak{g}$ not the Killing form of $\mathfrak{a}$?

I know that the Killing form of $\mathfrak{g}$ restricted to an ideal $I \subset \mathfrak{g}$ is just the Killing form of $I$. However, what happens in general if we relax the conditions and just ...
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Is the image of an ideal under a surjective Lie algebra homomorphism an ideal?

Say we have $\phi: \mathfrak{g} \to \mathfrak{h}$, where $\phi$ is a surjective Lie algebra homomorphism. Is $\phi(I)$ an ideal of $\mathfrak{h}$? I think this holds because for every element of $h \...
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How do we know a set can be ordered and how do we know how to form the order? Zorn Lemma

I was reading a lot of proofs that involves Zorn Lemma in Algebra. I give one example. All commutative ring $A$ with $1$ has at least one maximal ideal. The proof goes like this Set $\Sigma = \{...
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Ideals of $R[x]/I[x]$, where $I$ is a maximal ideal of $R$

Original Question: Let $R$ be a commutative ring with identity and $I$ maximal ideal in $R$. Show that $I[x]$ is a prime ideal in $R[x]$ and is not maximal ideal in $R[x]$, find two distinct maximal ...
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Finding a prime ideal whose class is order 10 in the ideal class group

"By factorising the ideal $(4+\sqrt{-74})_{R}$, or otherwise, find a prime ideal whose class $[P]$ in the ideal class group $Cl(R)$ has order 10" $R = \mathbb{Z}[\sqrt{-74}]$ So I've managed to get ...
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Divisorial ideal ($v$-ideal) of $A=\mathbb{Z}+X\mathbb{Q}[[X]]$.

I know that the domain $R$ is Mori if and only if for every nonzero ideal $I$ of $R$, $I_v=(a_1,\dots,a_n)_v$ for some $a_1,\dots,a_n\in I$. My goal is to find an example satisfying : For any $v$-...
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Contraction of quotient ideal is quotient of contractions?

Let $\mathfrak a,\mathfrak b$ be ideals in a ring $A.$ The quotient of $\mathfrak a$ and $\mathfrak b$ is $(\mathfrak a:\mathfrak b)=\{x\in A:x\mathfrak b\subseteq \mathfrak a\}$ and if $f:B\to A$ is ...
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Prove $(y-x^2)$ is a prime ideal in $\mathbb{R}[x,y]$, but not maximal.

My guess is to use the fact that when we take the quotient, $\mathbb{R}[x,y]/(y-x^2)$, this will become an integral domain but not a field. I am not sure how to take the quotient, though. I am also ...
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example of right / left ideals

I searched and searched for examples of right / left ideals, but could find none. I read that a right ideal of $S$ is a subset of $R$ of $S$ such that $RS \subseteq R$, and that symetrically, a left ...
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When do two ideals have trivial intersection?

Let $R$ be a ring and $I,J$ two ideals. If $I \cap J=0$ then $ij=0$ for every $i \in I$ and $j \in J$. This happens when $R=A \times B$ and $I=I’ \times \{0\}$ and $J=\{0\} \times J’$ with $I’$ ideal ...
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Radical of the ideal generated by $x^3 - y^6$ and $xy-y^3$

I am following Andreas Gathmann's notes on algebraic geometry. He asks, right after he shows that $V$ and $I$ are bijection between algebraic varieties and radical ideals (so I suppose I must use ...
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Check if the following ideals in $F[x, y]$ are closed ideal

[Definition] Let $F$ be a field. For $E \subseteq F^n$. The ideal of $E$, denoted $I(E)$, is $$I(E)=\left\{ f \in F[x_1, \cdots, x_n] : f(x)=0 \ \ \forall x \in E \right\}.$$ An ideal $I \subseteq ...
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coprime ideals in a ring

Suppose $R$ is a ring ($R$ may not have a unit and can be non-commutative), $I,J$ are two nonzero proper ideals in $R$ such that $I+J=R$ and $I\cap J\neq 0$. I wonder if there exists a possibility ...
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Finding generator for the ideal generated by $a$ and $b$ in an euclidean domain

Let $D$ be a euclidean domain and $a, b \in D$. Show that $M = \{xa + yb \ \mid \ x, y \in D\} $ is an ideal of $D$. Find $d \in D$ such that $M = \langle d \rangle$ and prove your claim. My effort: ...
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Coloring of hypergraph with polynomial implication

Sorry to bother again with my misunderstandings, but I encountered yet again an issue with the topic of coloring in graphs and again I require some help to deal with this exercise: Let $\mathbb{F}$ ...
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$\sqrt{180\mathbb Z} = 30\mathbb Z$, $(180\mathbb Z:700\mathbb Z)= 9\mathbb Z$

(First time studying rings, and I need some help on this example about radical ideal and fraction ideal) Let $\sqrt{I}$ be the radical ideal on the commutative ring $R$, defined as $\sqrt{I}=\{r\in R:...
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Image of an ideal under a surjective ring homomorphism is an ideal

Let $\phi: R \longrightarrow R'$ be a surjective ring homomorphism and $I$ an ideal in $R$. Show that $\phi(I) = \{ \phi (r) : r \in I \}$ is an ideal in $R'$. So I asked this question a couple ...
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Factorising the ideal $(14)$

I wish to find the prime factors of the ideal $(14)$ in $\mathbb{Q}(\sqrt{-10})$. My working so far has been by noticing that $$14=(2+\sqrt{-10})(2-\sqrt{-10})=2\times7$$ So we have the candidates $...
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Show that $x^2 +1$ is irreducible in $\mathbb{R}[x]$, but it has roots in $\mathbb{R}[x]\space/\space(x^2 +1) \cong \mathbb{C}$ [duplicate]

So I know that for something to be irreducible, then it cannot be written as the product of non-constant polynomials of smaller degree, but I don't know how to show that the factors don't exist is the ...
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If $\phi:R\rightarrow R'$ is a surjective ring homomorphism and I is an ideal in R… continued below [duplicate]

then $\phi(I)=[s'\in R'|s=\phi(s)\space \forall\space s\in I]$ is an ideal in R' So I know that for something to be an ideal, it needs to be closed under subtraction and it must absorb products. I ...
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Calculate generators of an intersection of Ideals

$$I = (x_1^2-x_1,x_2^2-x_2,...,x_n^2-x_n,t-\sum_{i=1}^n 2^{i-1}*x_i)$$ Ideal in $\mathbb Q[x_1, ..., x_n,t]$. How can I calculate the generators of $J = I \cap \mathbb Q[t]$ by hand? I tried it with ...
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Prove: If $\mathit R$ is a commutative ring with unity and $\mathit I=(x)\subseteq R[x]$, then $R[x] / (x)\cong R$ [duplicate]

I know that to show a ring is isomorphic to another ring, I have to find a bijective ring homomorphism between the two rings. Or I could use the F.H.T. but I would also need a function to make that ...