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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Understanding the definition of semisimple Lie algebras in terms of ideals

I'm struggling to understand the definition of a semisimple Lie algebra. The definitions I'm using are: Simple: "A Lie algebra $\mathfrak{g}$ is simple if it is non-abelian and contains no non-...
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Atiyah-Macdonald #7.19, about “irreducible decomposition” of ideal.

An ideal $\mathfrak{a}$ is said to be irreducible iff $\mathfrak{a}=\mathfrak{b}\cap \mathfrak{c} $ implies $\mathfrak{a}=\mathfrak{b} $ or $\mathfrak{a}=\mathfrak{c}$ The question is following. 7.19 ...
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Finding all elements in $I_{R/B}(A/B)$ and $(I_R(A))/B$ as following

Let $R$ be a ring and $A$ be a right ideal in $R$. Define $I_R(A) = \{r \in R | rA \subseteq A \}$. If $B$ be an ideal in $R$ and $B \subseteq A$, find all elements in $I_{R/B}(A/B)$ and $(I_R(A))/B$. ...
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Prove that $\mathbb{Q}[x,y]$ contains an ideal $I$ which can be generated by 3 elements, but not by 2 elements.

My first thought was $(2,x,y)$. But as in this post, Show that any ideal in $\mathbb{C}[x,y]$ containing $y$ can be generated by $2$ elements, $(2,x,y)$ would be generated by 2 elements. Does anyone ...
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$R$ is a subring of a field $F$ with $x\in R$ or $x^{-1}$ for all $x\in F^\times$. If $I,J$ are ideals of $R$, then $I\subseteq J$ or $J\subseteq I$. [duplicate]

Let $R$ be a subring of a field $F$ such that for all $x\in F^\times$, either $x\in R$ or $x^{-1}\in R$. Prove that if $I$ and $J$ are ideals of $R$, either $I\subseteq J$ or $J\subseteq I$. I am ...
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finding an annihilator of ideal in the ring $\mathbb{Z}_{20}$ as follows

Here's the complete problem. Let $I$ be an ideal of the commutative ring $R$. Define the annihilator of $I$ to be the set $Ann(I) = \lbrace r\in R \mid ra = 0 \ for \ all \ a \in I \rbrace$. Let $\...
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Associativity in Non commutative rings

I came across a paper, in which one of the axiom stated: $\forall x,y \in$ R, $L$ is a left ideal of R, we have $x(Ly)=(xL)y$ I am not able to see how this is true. Cause for the LHS we have right ...
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All element of the quotient ring $\mathbb{Z}_m/I$ for some $m \in \mathbb{N}$.

Let $I = \lbrace \overline{0}, \overline{8}, \overline{16} \rbrace$ be an ideal in $\mathbb{Z}_{24}$. Find all elements of quotient ring $\mathbb{Z}_{24}/I$. The answer is $\mathbb{Z}_{24}/I = \lbrace ...
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Proving that product of difference ideal (left-right ideal) is subset of their intersection as follows.

Let $R$ be an arbitrary ring, $I$ and $J$ be the right and left ideals, respectively. Prove that $IJ \subseteq I \cap J$. My attempt: Let $x \in IJ$. That is, $x = i_1j_1+i_2j_2+\dots + i_nj_n$, for ...
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Is the quotient ideal $JI^{-1}$ of two finitely generated fractional ideals $J,I$ finitely generated if both $I$ and $J$ contain a regular element?

Let $R$ be a noetherian ring without embedded primes (e.g. $R$ reduced). Let $I,J \subseteq \operatorname{Frac}(R)$ be two finitely generated $R$-submodules of $\operatorname{Frac}(R)$ both containing ...
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Superfluous right ideals of a formal triangular matrix ring.

Definition A right ideal $I$ of a ring $R$ is said to be superfluous (or small) if there is no proper right ideal $J$ of $R$ such that $I+J=R$. I am stuck in finding superfluous right ideals of a ...
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Examples of ideals of non Noetherian Lie algebra

Could someone give an example of ideals of non Noetherian Lie algebra, please? A Lie algebra $L$ satisfy the maximum condition for ideals, if for each , ascending chain $H_{1} \subseteq H_{2} \...
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Is $F[x]/q(x)=F[x]/\langle q(x)\rangle$?

Let, $F[x]$ is a ring of integer and $q(x) \in F[x]$. What is the difference between $F[x]/q(x)$ and $F[x]/\langle q(x)\rangle$? By definition $F[x]/q(x)$ is the set (quotient ring, thus additive ...
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Inverse image of anhilator ideals

Let $f$ be a ring homomorphism from $R\rightarrow S$ and $J$ be annihilator of some ideal in $S$. Under what conditions on $R$ and kernel of $f$ , $f^{-1} (J) $ is annihilator of some ideal of $R$. ...
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Ideal as an intersection of larger ideals

Let $I = (x(z-1),x^2-yz)$. I'm asked to prove that $I = (x,x^2-yz)\bigcap(z-1,x^2-yz)$. I was able to prove that $I$ is contained in this intersection, but couldn't figure out the other one: I could ...
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Is the finite intersection of prime ideals radical?

Does there exist a ring $R$ and finitely many prime ideals $P_i$ such that $\cap_{i = 1}^n P_i$ is not radical ideal? In other words, is the finite intersection of prime ideals a radical ideal?
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The ideal generated by $x^m-1$ and $x^n-1$ in $\mathbb{Z}[X]$ is principal. [duplicate]

I can show that $(x^m-1, x^n-1) \subseteq (x^{(m,n)} - 1)$, but I am stuck with the other inclusion. i.e. showing there exist polynomials $p, q \in \mathbb{Z}[X]$ such that $p(x)(x^m-1) + q(x)(x^n-1) =...
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Pre image of product of ideal

Let $f$ be a surjective homomorphism from $R$ to $S$. How pre image of product of ideal $f^{-1}(I_1...I_n)$and product of pre images of ideals $f^{-1}(I_1)...f^{-1}(I_n)$ are related. I know they ...
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Is $(p)$, $p$ prime, ideal prime in $\mathbb{Z}[x,y]$?

Is correct the following? I want verify if $(p)$ is prime in $\mathbb{Z}[x,y]$ ($p$ prime) I have this: $\mathbb{Z}[x,y]/(p)\simeq (\mathbb{Z}[x]/(p))[y]\simeq (\mathbb{Z}_{p}[x])[y]$ and $\mathbb{...
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Prove the zero set of a proper ideal of the ring of continuous complex-valued function on a compact space is nonempty

Prove the zero set of a proper ideal $I$ of the ring of continuous complex-valued function on a compact space $X$ is nonempty. The above problem is from Lang' s Real and Functional analysis Chapter ...
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Question about product of ideals in a $C^*$-algebra

Consider the following fragments from Murphy's book '$C^*$-algebras and operator theory' I'm trying to understand why $B \cap I = BIB$. Attempt: The inclusion $BIB \subseteq B\cap I$ is trivial since ...
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Quotient of ring by radical ideal.

Let $R$ be a commutative ring with identity, and let $I$ be an ideal of $R$. Let $J$ be the radical of $I.$ What can we say about the quotient ring $R/J?$ Does it have any special properties? In ...
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Smallest closed ideal containing an element in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a \in A$. I want to describe the smallest closed ideal containing $a$. If the algebra is unital, I think this ideal will be $\overline{AaA}$. But can we describe this ...
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Questions involving maximal ideals and cartesian product

Lets $A,B$ be two rings. Show that $M\times N$ is a maximal ideal of $A\times B \iff$ $M\times N$ is the form $I \times B$ or $A \times J$, where $I$ is a maximal ideal of $A$ and $J$ a is a maximal ...
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Prime ideals are maximal among principal ideals: geometry?

The claim is for a domain $R$, among principal ideals of the form $(r)$ for $r \in R$, the principal ideals which are prime are maximal among principal ideals. That is, we have $(p)$ a principal ideal ...
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How does the quotient ring $\Bbb Z[x]/(x^2-x,4x+2)$ look like?

How does the quotient ring $\Bbb Z[x]/(x^2-x,4x+2)$ look like? Normally to solve this you play around with generators until you get something you can work with. I was unsuccessful in reducing it to ...
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The definition of the resultant of two polynomials over a UFD

In the book Principle in Algebraic Geometry of Griffiths & Harris, I encountered the following definition of the resultant of two polynomials: If $R$ is a UFD and $u,v\in R[t]$ are relatively ...
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Show that $\mathfrak{m}_p$ is an ideal in $\mathcal{O}_V.$

I'm working through Algebraic Geometry: A Problem Solving Approach by Garrity et al, and I have found myself stuck on Exercise 4.13.1, which is the section Points and Local Rings. Let $V = V(x^2 + y^...
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Characterize ideals in a number ring

The following question is taken from D. A. Marcus' Number Fields Chapter 3 Exercise 9(c). There is a question on this already but I do not understand the answer there provided by @AdamHughes (...
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Find ideal, char and subring [closed]

Let S be nonempty set of elements. On partitive set P(S) we can define operation ∩ and A△B=A\B U B\A. If we know (P(S),△,∩ ) is ring and A⊂S. How ca I check if P(A) is subring or ideal of P(S)? And ...
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Showing that $q=(z_1,z_2^2)$ is primary in $\mathcal O_2 $

Show that if $q$ is primary, then $\sqrt{q}$ is prime. Show that in the ring $\mathcal O_2 = \mathbb C\{z_1, z_2\}$, $q=(z_1,z_2^2)$ is primary. original picture I already did the first item. For the ...
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How to show that $(3,x-1)\not=(3,x+1)$ as ideals in $Z[x]$

How to show that $(3,x-1)\not=(3,x+1)$ as ideals in $Z[x]$ Both are maximal. I think all i need to do is to show is that $x+1 \not \in (3,x-1)$ but I do not know how to show that. I wanted to assume ...
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How does the ring structure $10\mathbb{Z}/5\mathbb{Z}$ [closed]

Now the set of integers $10\mathbb{Z}=0,10,-10,20,-20,..$ and $5\mathbb{Z}=0,+5,-5,+10,-10,...$ .How does $10\mathbb{Z}/5\mathbb{Z}$ will look like -$10\mathbb{Z},5+10\mathbb{Z}$?
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Ideals $(X^2+1)$ and $(X^2+1, 7)$ of polynomial ring $\mathbb{Z}[X]$ [closed]

How can I show that generated ideals $(X^2+1)$ and $(X^2+1, 7)$ of polynomial ring $\mathbb{Z}[X]$ are a prime ideal and a maximal ideal, respectively?
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How do we interpret a zero sequence in the context of ideal theory?

We already know and may simply understand the definition of a zero sequence in $\mathbb{Q}$ - it is just a sequence, which converges towards $0$. Given the context of ideal theory, let $R$ be a ring ...
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Example of a ring with a unique two sided maximal ideal which is not a local ring (that is it has more than one left or right maximal ideals).

Let $R$ be a ring (possibly non-commutative). Definition $R$ is called a local ring if it has a unique left(and equivalently right) maximal ideal. I am looking for an example of a ring (obviously non-...
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How $\{am + pn : m, n \in \mathbb{Z}\}=\langle 1 \rangle$?

I don't understand how $\{am + pn : m, n \in \mathbb{Z}\}$ is equal $\langle 1 \rangle$, doesn't $\langle 1 \rangle$ contains all integer of $\mathbb{Z}$? the passage I got it from - Prime ideal ...
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How can I understand a hypocycloid as an ideal in a polynomial ring?

Today I was reading On teaching mathematics by V. I. Arnol’d and came across the following quote. "Rephrasing the famous words on the electron and atom, it can be said that a hypocycloid is as ...
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Exhibit the ideals of $\mathbb{Z}[x]/(2,x^3+1)$

I start by trying to see the homomorphism between $\mathbb{Z}[x]$ and $\mathbb{Z_2}[x]$.I define the homomorphism by $\phi:(ax^i)=a(mod 2)x^i$.It is trivial to see that it a homomorphism. The kernel ...
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$H$ normal iff $Lie(H)$ is an ideal.

I was reading a proof of the following theorem. Theorem 20.28 (Ideals and Normal Subgroups). Let $G$ be a connected Lie group, and suppose $H \subset G$ is a connected Lie subgroup. Then $H$ is a ...
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Prove $[\mathfrak g,\mathfrak g]$ is an ideal.

I have to show : Given a Lie algebra $\mathfrak g$, then $[\mathfrak g,\mathfrak g]$ is an ideal. I was told to use Jacobi's identity, but I am not sure why. It seems I just have to show that for $x,...
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Equivalency of the maximum and minimum conditions of idempotents of a ring

Let $R$ be a ring with unit, and let $I$ be the set of all idempotents of $R$, that is, all $e\in R$ such that $e^2 = e$. We put a partial ordering $\leq$ on $I$ by saying $e\leq f$ if $ef=e=fe$ or ...
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Why every field is a local ring, but the ring of integers $\mathbb{Z}$ not?

It is said, that $\mathbb{Z}$ is trivially not a local ring, because the sum of any two non-units must be a non-unit in a local ring and for example $-2+3=1$. But why every field is said to be local ...
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Properties of germs of varieties and ideals

proposition I'm trying to demonstrate this proposition, but I'm having some trouble with it, and the book says it comes straight from the definition. What I tried to do: Define $I_1=(f_{i_1}, ..., f_{...
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UFD is also an ideal of a ring

Is it true that when a UFD is another ring $R$'s ideal, then ring $R$ is also a UFD? I find an example but I'm not sure: the holomorphic ring $\mathcal O_x$, it's a UFD and the meromorphic ring $\...
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Prove an equality between ideals in $\mathbb{C}[x, y]$.

I have to prove that in $\mathbb{C}[x, y]$ we have the equality $$ (x^3-x^2, x^2y-x^2, xy-y, y^2-y)=(x^2, y) \cap (x-1, y-1).$$ I have proven the inclusion $\subset$ proving that every generator of ...
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Show that $I$ is an ideal of $\mathbb{K}[x]$

Let $\mathbb{K}$ be a field, $x_1, x_2, x_3\in \mathbb{K}$ and $I:=\left \{f\in \mathbb{K}[x]\mid f(x_1)=f(x_2)=f(x_3)=0\right \}$. I want to show that $I$ is an ideal of $\mathbb{K}[x]$. So we take $...
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Why is $I = \left \{ \sum_{i=0}^n a_i x^i | a_0 \in 2 \mathbb{Z} \right \} \subseteq \mathbb{Z}[x]$ not a principal ideal?

Why is $I = \left \{ \sum_{i=0}^n a_i x^i | a_0 \in 2 \mathbb{Z} \right \} \subseteq \mathbb{Z}[x]$ not a principal ideal? We saw it as a short example for a non-principal ideal in a linear algebra ...
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Let $I$ and $J$ be two ideals. Then is $IJ=JI$?

In a commutative ring $A$, $IJ=JI$ where $I$ and $J$ are two ideals of $A$. But if $A$ is not commutative is it true?? $IJ$ is the product of ideals $I$ and $J$.
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GCD of Ideal: How we get $\gcd(I, J) = I + J $?

Take any two non-zero ideals $I$ and $J$ in $R$. Since we know that ideals in a Dedekind domain factors uniquely into prime ideals $$I = \prod_i P_i^{m_i}, J = \prod_i P_i^{n_i}$$ where $P_i$’s are ...

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