Mathematics Stack Exchange
Mathematics Stack Exchange

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.

An ideal $I$ in a ring $(R,+,\cdot)$ a subset $I\subseteq R$ such that $(I,+)$ is a subgroup of the additive group $(R,+)$ and $r\cdot x,x\cdot r\in I$ whenever $r\in R$ and $x\in I$ (i.e., $I$ is closed under multiplication by arbitrary elements).

This is the most frequent use of the name ideal, but it is used in other areas of mathematics too:

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Unique prime ideal containing $(2)$ in $\mathbb{Z}[\sqrt{-3}]$

I'm having trouble with an algebraic number theory problem. Let $R = \mathbb{Z}[\sqrt{-3}]$. The problem is to show that $(2, 1 + \sqrt{-3})$ is the unique prime ideal containing the ideal $(2)$, and to conclude that $(2)$ can't be written as a…
Ethan Alwaise
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For a ring R, and ideals $A$, $B$, then $AB=A \cap B$ if $A + B = R$

$AB \subseteq A \cap B$ is clear. I have seen reverse inclusion proven thus, Let $x \in A\cap B$. Since $A+B=R$, there exist $a \in A$, $b \in B$, such that $a+b=1$. Then $x= axa + axb + bxa + bxb$. Therefore, $A \cap B \subseteq AB$. My problem: I…
Rachmaninoff
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Non Units of Commutative Ring All Being Contained in Some Ideal M which is not R

Let R be a commutative ring with $1_R$. Then I'd like to prove the equivalence between those statements, particularly the point from $(ii) \to (iii)$ : (i) all non units of R are contained in some ideal M $\ne R$ (ii) the set of all non units of…
Beverlie
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Ideal gen. by a set S = Intersection over ideals containing S

I am trying to prove the following statement: Let R be a ring and $I=\{\sum_{i=1}^n a_i x_i : a_i\in R\}$ the ideal generated by $S=\{x_1,\ldots, x_n\}$. Then $I$ is the intersection of ideals $J$ in R containing $S$. Let $a=\sum_{i=1}^n a_i x_i\in…
Phil-ZXX
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In which way are sigma ideals a special case of ideals?

The article on sigma-ideals in wikipedia claims they are a special kind of ideals: http://en.wikipedia.org/wiki/Sigma-ideal But, unfortunately, no explanation to that regard is offered (not at least that I can identify as such). Could anyone explain…
Javier Arias
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Ideal in ring product

Let $R$ and $S$ be two rings, then $K$ is an ideal in $R\times S$ if and only if there are $K_1$ and $K_2$ such that $K_1$ is an ideal of $R$ and $K_2$ is an ideal of $S$ such that $K = K_1 \times K_2$. How do you show that $K_1 = \{a \in R \mid…
Newbie Math
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Is every non-trivial ideal in a commutative ring is a principal ideal?

I'm a bit lost... it seems every non-trivial ideal in a commutative ring is a principal ideal. but is it true? if not, could you pls give a counter example?
athos
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Intersection and Sum of Polynomial Ideals from different rings

It is well known that intersection and sum of polynomial ideals from the same ring are lattice operations. I wonder if this is still true for ideals from different rings (over the same field). Specifically, let I be an ideal over the ring k[x,y],…
Tegiri Nenashi
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If $I \cap J \cap K = IJK$ for proper ideals $I$, $J$, and $K$ (not containing each other) then does $I \cap J = IJ$?

Let $R$ be a commutative Noetherian ring. Then, I cannot construct a counterexample (or proof) for the question posed in the title. If $R$ is a polynomial ring and all ideals involved are monomial ideals, then this statement is true since it will…
Rellek
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Ideal of K[x,y], need for some precisions.

I was reading this topic to show that $(x-a,y-b)$ is an ideal of $K[x,y]$, where $K$ is a field. The answer suggests to show that $(x-a,y-b)$ is the kernel of the evaluation in $ev_{(a,b)} : K[x,y] \rightarrow K$. I understand that…
roi_saumon
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Algebraic number theory: Factor into prime ideals

I have been working through a book on Ring/Algberaic number theory recently and the book only has solutions for even questions. I was wondering if somebody could help with a solution for the following question. I am so far unsure, the route I have…
RedPvwer
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Let $A$ and $B$ be two ideals of a commutative ring $R$ with unity such that $A+B=R$. Show that $AB = A \cap B$

I have proved one part: $AB \subset (A \cap B) \quad ...(1)$ To prove second part $(A \cap B) \subset AB$ my approach is as following: Let $x \in A \cap B \Rightarrow x \in A \text{ and } x \in B \quad ...(2)$ $\because A$ and $B$ are ideals of ring…
abhinav12369
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Finding generators for products of ideals

If you want to find the generators for the product of ideals, do you simply take all possible products of the generators in the ideals. For example, let $R$ be a ring and let $I = (a,b)$ and $J = (c,d)$. Then we…
Ethan Alwaise
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Relation between ideal numbers and ideals of a ring?

I would like to know whether the ideal numbers of Kummer (or the ideals of Dedekind for that matter) are closely related to the concept of ideal (right ideal, left ideal, two-sided ideal and so on.....) in a ring. If so, in which manner? Or are they…
Javier Arias
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how to identify the ideals of a ring by using canonical homomorphism?

Assume we have a quotient ring $R'=\mathbb{C}[t]/(t-1) $. How can I find the ideals of $ R' $ by using the cannonical homomorphism $ H$ from $\mathbb{C}[t] $ to $ R' $. This is my homework actually but since I want to deal with it my self I quite…
Ulgen
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