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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What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
29
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6answers
2k views

Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong ...
26
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3answers
698 views

Ideals of $\mathbb{Z}[X]$

Is it possible to classify all ideals of $\mathbb{Z}[X]$? By this I mean a preferably short enumerable list which contains every ideal exactly once, preferably specified by generators. The prime ...
22
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4answers
632 views

Why do we study prime ideals?

I hope this isn't an inappropriate question here! I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? ...
16
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8answers
1k views

Intuition behind “ideal”

To briefly put forward my question, can anyone beautifully explain me in your own view, what was the main intuition behind inventing the ideal of a ring? I want a clarified explanations in these ...
16
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3answers
536 views

If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent

I'm solving a problem from Atiyah-Macdonald. I have to show that if $X=\mathop{\mathrm{Spec}}A$ is not connected then $A$ contains idempotents $e \neq 0,1$. The converse is easy. If $e \in A$ ...
15
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0answers
238 views

Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
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3answers
1k views

A maximal ideal is always a prime ideal?

A maximal ideal is always a prime ideal, and the quotient ring is always a field. In general, not all prime ideals are maximal. 1 In $2\mathbb{Z}$, $4 \mathbb{Z} $ is a maximal ideal. ...
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2answers
2k views

What do prime ideals in $k[x,y]$ look like?

Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like? As far as I know, the maximal ideals of $k[x,y]$ are of the form ...
12
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1answer
125 views

Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$

Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$. I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
11
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6answers
3k views

Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$

Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is ...
11
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5answers
474 views

How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
11
votes
3answers
310 views

What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?

I am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number ...
11
votes
3answers
910 views

Complement of maximal multiplicative set is a prime ideal

Let $R$ be a commutative ring with identity. I've been trying to prove the following: If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$. I have spent ...
11
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3answers
533 views

Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$

In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization: A lattice is distributive if and only if ...
10
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3answers
423 views

Simple example of non-arithmetic ring

Can anyone provide a simple concrete example of a non-arithmetic commutative and unitary ring (i.e., a commutative and unitary ring in which the lattice of ideals is non-distributive)?
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2answers
173 views

If every ascending chain of primary ideals in $R$ stabilizes, is $R$ a Noetherian ring?

A commutative ring $R$ is called Noetherian if every ascending chain of ideals in $R$ stabilizes, that is, $$ I_1\subseteq I_2\subseteq I_3\subseteq\cdots $$ implies the existence of $n\in\mathbb{N}$ ...
9
votes
2answers
890 views

Methods to check if an ideal of a polynomial ring is prime or at least radical

I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example. To be more explicit, I am working ...
9
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1answer
364 views

Every maximal ideal is principal. Is $R$ principal?

Let $R$ be a commutative ring with 1. If every maximal ideal of $R$ is principal, is $R$ a principal ring?
9
votes
1answer
225 views

Conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$

Let $A$ be a commutative ring with identity and, $\mathfrak{a}$ and $\mathfrak{b}$ ideals.I'm trying to find sufficient and necessary conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + ...
9
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1answer
170 views

Isomorphic factor rings of polynomial rings does imply isomorphic ideals?

Let $k$ be a field, $I$ and $J$ are ideals of $R=k[x_1,\dots,x_n]$. If $R/I\simeq R/J$ as rings, then $I \simeq J$ as $R$-modules holds? Thanks in advance!
9
votes
3answers
227 views

Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.

Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime. My attempt: Let $I$ be an ideal of $R$. Then we have $I$ is maximal $\Leftrightarrow$ $R/I$ is a finite ...
9
votes
2answers
350 views

Intersection of finitely generated ideals

Let $I$, $J$ be finitely generated ideals in a ring $A$ (commutative with identity). I know that the intersection need not be finitely generated: can somebody give me an example? Thanks.
9
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1answer
222 views

A ring that has exactly 7 left ideals (T. Y. Lam)

Exercise 3.25 in Lam's First Course states: Let $R$ be a ring that has exactly seven nonzero left ideals. Prove that one of them is an ideal (i.e. left and right) and provide an example of such a ...
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1answer
178 views

Ideal in compact Hausdorff space

This is exercise 70, chapter 4. from Folland (page 142) Let $X$ be a compact Hausdorff space. An ideal in $C(X, \mathbb{R})$ is a subalgebra $J$ of $C(X, \mathbb{R})$ such that if $f\in J$ and $g\in ...
9
votes
0answers
256 views

Maximal ideal space of $C^*$-algebra of Riemann integrable functions

Let $R([0,1])$ be the unital commutative $C^*$-algebra of complex valued Riemann integrable functions on $[0,1]$ with pointwise operations and the supremum norm. In the 1980 paper The Gelfand space ...
8
votes
5answers
812 views

If the localization of a ring $R$ at every prime ideal is an integral domain, must $R$ be an integral domain?

Let $R$ be a commutative ring. Suppose that for every prime ideal $p$ of $R$, the localized ring $R_p$ is an integral domain. Must $R$ be a integral domain? I was trying to think of counter-examples, ...
8
votes
1answer
205 views

If $I$ is a finitely generated ideal of $A[X]$, is $I\cap A$ necessarily finitely generated for a commutative unital ring $A$?

Let $A$ be a commutative ring with $1$ and $A[X]$ the ring of polynomials in one variable over $A$. Assume $I$ is a finitely generated ideal of $A[X]$. My question is Is $I\cap A$ necessarily ...
8
votes
1answer
184 views

Noetherian ring whose ideals have arbitrarily large number of generators

Does a commutative ring satisfying the following two properties exist? All ideals are finitely generated; There are prime ideals with arbitrarily large (finite) minimal generating sets.
8
votes
2answers
459 views

Infinite product of fields

The main source of inspiration for this question is this excerpt Recall: An ultrafilter on the set X gives you a maximal ideal in the ring of all real-valued functions, and these are the only ...
8
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2answers
70 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
8
votes
2answers
141 views

Counterexample: multiplying modules by elements of an ideal vs. taking linear combinations

Let $R$ be a ring (commutative, unital) and $M$ an $R$-module. Let $I \subset R$ be an ideal. We make the following definitions: $$ A := \{ am \ | \ a \in I,\ m \in M \} $$ $$ B := \left\{ ...
8
votes
2answers
1k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
8
votes
1answer
140 views

$I\cap J = P$ prime ideal, then $P=I$ or $P=J$

Question: Prove that if $I,J$ are ideals and $I\cap J=P$ is a prime ideal, then either $P=I$ or $P=J$. My proof: Suppose $P\ne I$. Then $I\cap J=P\subsetneq I$ and $\exists i\in I\setminus P$. Now ...
7
votes
7answers
587 views

Can someone explain ideals to me?

I'm struggling with the idea of ideals (both the definitions and the common notation). I'm in a basic collegiate algebra course, just looking for a bit of help. As simply defined as possible, if you ...
7
votes
2answers
142 views

Countable rings

Suppose we are given a countable unital ring $R$ with uncountably many distinct right ideals. Does it follow from this that $R$ has uncountably many maximal right ideals?
7
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3answers
493 views

Is $\lhd$ common notation for “is an ideal of”?

This question is because of this comment. I would like to know if I should refrain from using $\lhd$ for "is an ideal of" in ring-theoretic questions. Is it common enough, or should I explain what it ...
7
votes
3answers
300 views

Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
7
votes
4answers
330 views

Spectrum of $\mathbb{Z}[x]$

Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be ...
7
votes
1answer
115 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
7
votes
2answers
334 views

Are ideals in rings and lattices related?

There are (at least) two notions of ideals: An ideal in a ring is a set closed under addition and multiplication by arbitrary element. An ideal in a lattice is a set closed under taking smaller ...
7
votes
1answer
453 views

Maximal ideals in $R[x]$

I want to prove the following result: Let $R$ be a ring and $M$ a maximal ideal in $R$. If $P$ is a prime ideal in $R[x]$ that (strictly) contains $M[x]$, then $P$ is a maximal ideal in $R[x]$. ...
7
votes
3answers
67 views

Let $R = \mathbb Z[i]$. Show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$.

Let $R = \mathbb Z[i]$, $z = 3+i$ and $I = \langle z \rangle$. I need to show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$ and $10 \mid a$, ...
7
votes
1answer
129 views

Ideal of ideal needs not to be an ideal

Suppose I is an ideal of a ring R and J is an ideal of I, is there any counter example showing J need not to be an ideal of R? The hint given in the book is to consider polynomial ring with ...
7
votes
1answer
152 views

Prime elements in $\mathbb{Q}[[X,Y,Z]]$ whose status as an infinite series is unchanged by arbitrary multiplication

Let's suppose $R$ is the ring $\mathbb{Q}[[X,Y,Z]]$. I'm interested in finding power series $f(x,y,z) \in R \setminus \mathbb{Q}[X,Y,Z]$ which are, first of all, prime elements in $R$, but also ...
7
votes
1answer
148 views

Proper ideals generated by central ideals

Let $R$ be a unital ring and denote its center by $Z(R)$. If $I$ is an ideal of $Z(R)$, then the set $RI$ (consisting of finite sums of elements of the form ra where $r\in R$ and $a\in I$) is clearly ...
7
votes
2answers
95 views

Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
7
votes
2answers
301 views

Extension and contraction of ideals in polynomial rings

Suppose $I$ is an ideal in a polynomial ring $R=k[x,y]$. Let $\overline{k}$ be the algebraic closure of $k$ and let $S=\overline{k} [x,y]$. Then is $IS\cap R=I$?
7
votes
1answer
96 views

Ideal with large Grobner basis with respect to one monomial order

What is an example of a set of at most four polynomials $f_1,\ldots,f_n$ (in any number of variables) such that $\{f_i\}$ is a Grobner basis of $I=\langle f_i\rangle$ with respect to one monomial ...
6
votes
4answers
156 views

Let $R$ be a commutative ring with $1$. Suppose that every nonzero proper ideal of $R$ is maximal. Prove that there are at most two such ideals.

Let $R$ be a commutative ring with $1$. Suppose that every nonzero proper ideal of $R$ is maximal. Prove that there are at most two such ideals. Help me some hints. I have no idea to start. ...