# 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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### 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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### 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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### 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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### 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 ...
### 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 $J$.
### 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 ...