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, ...

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How do I compute the norm of a non-principal ideal of the ring of integers of a quadratic field without using ''large'' results

I am trying to compute the norm of the ideal $I=(7, 1+\sqrt{15}) \trianglelefteq \mathbb Z[\sqrt{15}],$ the ring of integers of $\mathbb Q[\sqrt{15}].$ I knew $I^2$ would be principal, as $I\bar ...
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prove $(m) \subset (n)$ iif $n$ divides $m$

For non-zero integers $m$ and $n$, prove $(m) \subset (n)$ iif $n$ divides $m$, where $(n)$ is the principal ideal. My attempt is following. For non-zero integers $m$ and $n$, assume that $(m) ...
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Relation between Noether's one-sided ideals and Polish notation?

Given the definitions of one-sided ideals (right ideals; left ideals) bu Emmy Noether, as referred in this answer Noether's definition of right and left ideals?, I would like to raise the ...
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Origin of ideals in order theory

I am trying to clarify in my head the different meanings of "ideals" in mathematics. We have ideals in Number Theory, as in Dedekind (derived from 'Ideal Complex Numbers' in Kummer), in Abstract ...
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A question on Join homomorphism and Ideals

On page 287 of the book Mathematical Methods in Linguistcs, by Barbara Partee, Alice Ter Meulen and Robert E. Wall (Dordrecht, Kluwer Academic Press, 1993), I find the following theorem, which they ...
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+50

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
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Visual representation of (right or left) ideals of a ring?

I would like to know about the canonical visual representation for ideals in a ring. Particularly, for the two kinds of one-sided ideals,that is, right ideals and left ideals. Is it possible to use ...
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32 views

Decision between right and left ideals of a ring?

Let us suppose whe find a phenomenon (in nature, social sciences, whatever) for which we believe (or some author has stated) it is possible a formalization or modeling in terms of ideal of a ring. Let ...
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53 views

Class group of $\mathbb Q(\sqrt{-55})$ and finding representatives for ideal classes

My first step in computing the class group of $\mathbb Q(\sqrt{-55})$ was to compute the Minkowski bound. Initially, I said $\lambda(-55)=2\sqrt{-55}/\pi<2(8)/3<6$ and I went the normal way of ...
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21 views

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 ...
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30 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
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47 views

Is the minimal number of generators of an ideal the rank of the ideal as a free $\mathbb Z$-module?

In an algebraic number theory course, my lecturer said that any ideal of $\mathcal O_K$, where $K$ is a quadratic number field, is generated by at most two elements. I am wondering why this is. When ...
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79 views

Question about ideals of a ring: $I\cdot J=I \implies J=I$?

Doing exercises, this question came to my mind. Is it true that if $I$ and $J$ are proper and nonzero ideals of a ring $R$, $$I\cdot J=I \implies J=I?$$ And $$I\cdot J=I \iff J\subseteq I?$$
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Residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$

What are the residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$? After finding the maximal ideals of $\mathbb{R}[X,Y]$, which are of the form: $\langle X-a,Y-b \rangle$ with $a,b ...
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64 views

Problem about ideals of the localization of a ring

I'm having problems on doing the section (ii) of this exercise. Let $R$ be a domain. Let $P$ be a prime ideal of $R$. (i) Prove that $S:=R\setminus P$ is a multiplicatively closed system with no ...
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51 views

Direct sum of non-zero ideals over an integral domain

Let $R$ be an integral domain. Let $I$ and $J$ be non-zero ideals of $R$. Is this statement always true: $$R\oplus(I\cap J)\cong I\oplus J\ ?$$ I regarded the short exact sequence $0\to I\cap ...
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34 views

$K[x,y]$ (where $K$ is a field) have any bound for the number of generators of ideals?

We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators. But is there any bound for the number of generators of arbitrary ideals? (For example in $K[x,y]$.)
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If $1\in M$, could $M$ be a maximal ideal of a commutative ring with identity $R$?

If $1\in M$, could $M$ be a maximal ideal of a commutative ring with identity $R$? I know that this is a very silly question. I think that the answer is that $M$ can't contain the identity for ...
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24 views

properties of finitely generated torsion-modules and their submodules over a PID

Let $R$ be a principal ideal domain, $M$ a finitely generated $R$-torsion module, and $N \subseteq M$ a submodule. I want to show that there exist free R-modules $F, F', F''$ and module homomorphisms ...
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358 views

infinitely many ideals

does the ring $\Bbb Z_2[x]$ have infinitely many ideals like $\Bbb Z[x]$? How do you know if a ring has a finite number of ideal. particularly asking about seemingly large rings.
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Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Let $R$ be a commutative ring with unity , I know that assuming axiom of choice , if $A$ is the set of all zero divisors (including $0$ ) then it is a union of prime ideals so it contains a prime ...
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24 views

Ideal in ring product [closed]

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 ...
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Maximal left ideals $\leftrightarrow$ simple left modules

Suppose $R$ is a ring with unity. This passage in Lang's Algebra discusses the correspondence $$\text{Maximal left ideals of $R$} \leftrightarrow \text{Simple left $R$ modules},$$ where I corresponds ...
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Question on intersection of ideals.

Consider the polynomial ring $k[x_0,x_1]$, and the two ideals $I=(x_0,x^2_0 x^2_1,x^3_1)$ and $J=(x^2_0,x^2_1)$. What is the intersection of these ideals? I found that $I \cap J = ...
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36 views

bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$

Given a ring $R$, I want to show that the localization of $R$ at the prime ideal $P$ of $R$(denoted as $R_P$) is isomorphic to the set of prime ideals of $R$ contained in $P$. That is: ...
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40 views

Finding primary decompositions of ideals

I have been given this example of the decomposition of an ideal into primary ideals $$ I =⟨x^2,xy,x^2z^2,yz^2⟩$$ Then the primary decomposition of this ideal is: $$⟨x^2,y⟩∩⟨x,z^2⟩⊆K[x,y,z]$$ This ...
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17 views

Irreducible decomposition of varieties vs primary decomposition of ideals

I'm new to working with varieties, and the statement mentioned below is left as an exercise, but I'm having some difficulty trying to prove it. Let $R=K[x_1,...,x_n]$. If $X=X_1\cup ... \cup X_n$, ...
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32 views

Show that the variety $V(I(X))=X$

In the ring $R=K[x_1,...,x_n]$, the variety of an ideal is defined as $V(I)=\{(a_1,...,a_n)\in K^n|f(a_1,...,a_n)=0, \space\forall f\in I\}$ The ideal of a variety is defined as $I(V)=\{f\in ...
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Questions about homomorphisms?

I'm running into algebra using homomorphisms for A LOT of things, but I don't think I have a full understanding of what they are. I have read on this site a good explanation that was really great, but ...
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39 views

Question about principle ideals and polynomials and quotient ring construction?

Say I have a ring of polynomials in $R[x]$. I wish to define the quotient group $R[x]/<x^2+1>$. My question lies in the ideal generated by $<x^2 + 1>$. This is the set of all numbers such ...
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28 views

Question about maximal ideals in a Polynomial ring

I'm reading Freiligh and he has an example in a book, here it is: Let $F = R$ and let $f(x) = x^2 + 1$. Which is well known to have no zeros in $R$ and thus is irreducible over $R$ by a theorem ...
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1answer
38 views

Solving Mordell Equations

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
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1answer
38 views

For a group ring, finding if a subset is an ideal. [closed]

For the ring $R=SG$, the group ring of a finite group G over an integral domain S, and a subset $I=(g-1|g \in G)$, is this subset an ideal? Is it prime? How about maximal?
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Showing a Ring and an Ideal are equal

Q: Let R be a commutative ring with unity. Prove that if A is an ideal of R and A contains a unit, then A=R. This is my attempt at an answer: It suffices to show that all the elements in R are in A. ...
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1answer
26 views

Proving that the three statements on units are equivalent

Let $R$ be a ring and $x\in R$ i) $x$ is a unit ii) $\bar{x}$ is a unit in $R/P$, where $P$ is a prime ideal iii) $\frac{x}{1}$ is a unit in $R_P$, where $P$ is a prime ideal i)$\implies$ ii) if $x$ ...
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1answer
32 views

how to tell if a ring is noetherian

In general how would i tell if the rings $\mathbb{Z}[\sqrt d]$ and $\mathbb{Z}[\frac{x}{y}]$ are noetherian? I know that the ring $\mathbb{Z}$ is noetherian as all ideals are contained in a finite ...
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1answer
59 views

Ideals in $\mathbb{Z}[X]$ with three generators (and not with two) [duplicate]

It is well-known that in $\mathbb{Z}[X]$ we do have non-principal ideals, for example $(2,x)$. This is an ideal with two generators. Now I was wondering if there exists an ideal with three generators, ...
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Ideals in $\mathbb{C}[x,y]/I$ where $V(I)$ is finite set.

$R = \mathbb{C}[x,y]$, $I \subset R$ ideal and $V(I)$ - is finite set. I want to prove that all prime ideals in $R/I$ are maximal. I know that it's true for $R/radI$. Let $\pi$ be natural map $R/I ...
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1answer
47 views

Showing an ideal is irreducible

I am currently trying to show that the ideal $\langle x^3, y^5, z^2 \rangle \subset \mathbb{C}[x,y,z]$ is irreducible (i.e.: it cannot be written as the intersection of two larger ideals $J$ and ...
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35 views

Rings, ideals and quotient rings

Suppose $I$ is an ideal of ring $R$, and $J'$ is an ideal of $R/I$. Show there is an ideal $J$ in $R$ so that $J/I=J'$. How do I answer this? What am I required to prove?
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Trouble finding the norm of the two following ideals

Given that $\alpha$ is the root of the polynomial, $x^3 - x - 1$ is $\alpha$ and $K=\mathbb{Q}(\alpha)$, show that the norm of the ideal $\langle 5, \alpha-2\rangle$ is $5$ and the norm of the ideal ...
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1answer
52 views

Varieties and ideals

I'm doing the exercises from Fulton of Algebraic Geometry and I'm stuck in the problem 2.44 Let $V$ be a variety in $\mathbb{A}^{n}$, $I=I(V)\subset k[x_{1},\ldots,x_{n}]$, $P\in V$ and let $J$ be ...
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1answer
30 views

Intersection and isomorphism of two relatively prime ideals

Let $R$ be a commutative ring. Using the definition that two ideals $I, J \subseteq R$ are relatively prime if $I + J = R$. I want to show that for two relatively prime ideals $I, J \subseteq R$, it ...
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31 views

How to show an ideal is principal

Is there a general procedure to check whether or not a prime ideal of the ring of integers $O_K$ is principal. In my case $K$ is a quadratic field, i.e $\mathbb{Q}(\sqrt {d})$, with $d$ square-free.
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For a maximal left ideal $M$ of $S$, is $f^{-1}(M)$ a maximal left ideal of $R$ when $f$ is surjective?

Let $f:R \longrightarrow S$ a surjective ring homomorphism. Is the inverse image $f^{-1}(M)$ a maximal left ideal of $R$ for any maximal left ideal $M$ of $S$? Comments: I tied something ...
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Why does $N(\mathfrak p)$ belong to $\mathfrak p$?

Why does $N(\mathfrak p)$ belong to $\mathfrak p$ ? $N(\mathfrak p)$ is the norm of the prime ideal $\mathfrak p\in\mathcal O$ defined as $N(\mathfrak p)=|\mathcal O/\mathfrak p|=$(say $p$) Now ...
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Question about maximal ideals?

I'm reading Freligh and introduction to Abstract algebra and I'm getting confused. The set generated by $\langle x^2 + 1\rangle$ is a maximal ideal in $R[x]$. First, I don't understand it. $\langle ...
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1answer
35 views

subring of a quotient field

Let $R$ be a principal ideal domain, and $S \subseteq Q(R)$ a subring of the quotient field of $R$, so that $R \subseteq S$. I want to show that, for any $x, y \in R$: $$\frac{x}{y} \in S \implies ...
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1answer
24 views

Why $\mathfrak p_2\cdots\mathfrak p_r\not\subset (a)\mathcal O$

If $(a)\mathcal O\subset\mathfrak p_1$ and $r$ is the minimal number such that $\mathfrak p_1\cdot\mathfrak p_2\cdots\mathfrak p_r\subset (a)\mathcal O$ then $\mathfrak p_2\cdots\mathfrak ...
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20 views

Proof that every Principal ideal domain is Noetherian [duplicate]

I would like to know if my logic is sound. We know that in every principal ideal domain, every ideal is multiplicatively generated. Thus, for $a \in R$ we have: $aR = ${$ra: r \in R$} Thus every ...