# 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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### Minimal non-zero prime ideal in a UFD

I'm trying to understand the following Proposition: Proposition: Let $R$ be a UFD (unique factorization domain) with subset $P$ of the set of irreducible elements of $R$. Then $P$ is in bijective ...
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### Construction of an ideal that is minimally generated by n elements

The following question is part of my abstract algebra assignment and I am looking for verification of solution. Question: Let A =k[x,y], where k is a field. For every $n\in \mathbb{N}$ , show that ...
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### Corollary of Weyl's Theorem - direct sum of simple ideals

Weyl's Theorem states that if $L$ is a semisimple Lie algebra over $\mathbb{C}$, then any finite dimensional representation of $L$ is completely reducible. I want to show that a corollary of this is ...
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### $lAnn_R (I)$ intersects nontrivially any non-zero right ideal of $R$

Let $I$ be a nilpotent ideal of ring $R$ (with unit). Then $lAnn_R (I)$ has a nontrivial intersection with any non-zero right ideal of $R$,where $$lAnn_R(I)=\{ r\in R\; |\; rI=0\}.$$ I was wondering ...
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### Given $K/F$ where $K$ is a field extension of $F$ of the form $F[x]/\langle p(x) \rangle$, what is the structure of $K[x]$?

$F$ is a field. $\langle p(x) \rangle$ is a maximal ideal. So $K = F[x]/\langle p(x) \rangle$ is a field extension. I am trying to understand what would be the structure of $K[x]$? $F[x]$ has all ...
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### How to approach proof of finding number of prime ideals lying over another prime ideal

Consider the following problem: Problem: Let $f= f(X) \in \mathbb{Z}[X]$ be a monic polynomial of positive degree and let $B=\mathbb{Z} [X] /\left<f\right>$. For a prime number $p$, show that ...
1 vote
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### Can the decomposition of a principal ideal in a Dedekind domain contain a non-principal ideal as a factor?

If $I= \prod_{P_i\in Spec(R)}P_i$ is the decomposition into prime ideals of a principal ideal $I$ in a Dedekind domain $R$, can one of the $P_i$ be a non-principal ideal? I guess it can’t, but I don’t ...
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### Multiplicity of intersection in terms of degree of the quotient of ideal by its saturation

In this guide to Macaulay2 I found an interesting way to compute intersection multiplicity. On page 61 the authors give an explicit case of the following argument: We want to compute the intersection ...
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### How to Prove an Ideal Can Be Generated From 2 Elements [duplicate]

Given a commutative ring with unity, $R$, an $a,b \in R$, and an ideal $I$ such that $I=\{ax+by \mid x,y \in R\}$. Prove that $I=(a,b)$. I think I want to show that $R$ is a PID, but I am not quite ...
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### Localization an ideal by maximal ideal [duplicate]

I try to show this, suppose $R$ be a commutative ring with $1$ and $A$ be an ideal in $R$. Show if $A_{M}=0$ for all maximal ideal $M$, then $A=0$. My idea this, If $a\in A$ then for every maximal ...
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### If factor ring is domain ( non commutative ), then quotient must be prime

Let $\frac{R}{P}$ be a domain ( not necessarily commutative ). Show that $P$ is prime. In previous questions asked on this site, integral domain ( commutative ) is assumed, but I don't think this is ...
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### Nilpotent ideals of algebra over a field.

I have 2 questions about the nilpotent ideals of an algebra $A$ over a field $K$. Each non-nilpotent left ideal of $A$ contains a nonzero idempotent element of $A$. The sum of all left nilpotent ...
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### Let $k$ be a field. Can all the ideals in $k[X,Y]$ be generated by at most two elements? [duplicate]

Let $k$ be a field. Can all the ideals in $k[X,Y]$ be generated by at most two elements? I don't think it can. I thought of the ideal $(X^2,XY,Y^2)$ and I think it can't be generated by only two ...
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### Prime ideal $\implies$ maximal in a Boolean ring

I want to show that a prime ideal in a non-unital Boolean ring $B$ is maximal ideal. If the ring contains unity then it is easy. As Boolean rings are commutative, for a prime ideal $P$ the ring $B/P$ ...
### Describe the ring structure quotient ring $Z[x,y]/(x^2,y^2,2)$.
This is taken from an exercise problem in Dummit and Foote's: Describe briefly the ring structure of $\mathbb Z[x,y]/(x^2,y^2,2)$ and show that $\alpha^2=1$ or $0$ for every $\alpha$ in the ring. I ...
Let $R$ be a ring with identity. If subrings are assumed to contain identity, does there exist an ideal $I$ and a subring $S$ of $R$, where $I\cap S$ is not an ideal of $R$?