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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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 ...
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4answers
500 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? ...
13
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0answers
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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
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
12
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8answers
668 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 ...
11
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2answers
156 views
If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent
I'm solving a problem from Atyah-Macdonald. I have to show that if $X=\mathop{\mathrm{Spec}}A$ is not connected then $A$ contains idempotent $e \neq 0,1$. The converse is easy. If $e \in A$ is an ...
11
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1answer
79 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 ...
10
votes
3answers
349 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)?
10
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3answers
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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 ...
9
votes
5answers
328 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 ...
9
votes
2answers
160 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
84 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 ...
8
votes
2answers
521 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 ...
8
votes
1answer
118 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
2answers
319 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
votes
1answer
141 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
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3answers
381 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 ...
8
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1answer
68 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 ...
8
votes
0answers
218 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 ...
7
votes
3answers
629 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. ...
7
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2answers
124 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
171 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
2answers
690 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 ...
7
votes
2answers
141 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.
7
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1answer
135 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 ...
6
votes
4answers
205 views
When an ideal is a prime ideal
Let $I$ be the subset of $\mathbb{Q}[x]$ that consists of all the
polynomials whose first five terms are 0.
I've proven that I is an ideal (any polynomial multiplied by a polynomial in $I$ must ...
6
votes
3answers
312 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 ...
6
votes
4answers
198 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 ...
6
votes
4answers
248 views
$I$ is maximal $\implies I$ is prime
Been asked to show this is true with hints $R/I$ field $\Longleftrightarrow$ $I$ is maximal and $R/I$ integral domain $\Longleftrightarrow$ $I$ prime.
Can you check this please, I have had a ten ...
6
votes
3answers
121 views
For $I,J$ ideals, show that $IJ\subseteq (I\cap J)(I+J)$
For $I,J$ ideals, show that $IJ\subseteq (I\cap J)(I+J)$. If it helps $I$ and $J$ are ideals in a Dedekind domain, but as far as I can tell the proof given in the book only uses the fact that we are ...
6
votes
4answers
131 views
Finding the ideals in a ring of fractions
I am dealing with the ring $$R=\left\{\frac{a}{b} \mid a,b\in\mathbb{Z}\mbox{, $b$ is not divisible by 3}\right\}$$ with addition and multiplication as defined in $\mathbb{Q}$ and I'm trying to find ...
6
votes
1answer
86 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 ...
6
votes
1answer
131 views
Question about proof of Going-down theorem
I have written a proof of the Going-down theorem that doesn't use some of the assumptions so it's false but I can't find the mistake. Can you tell where it's wrong?
*Going-down*$^\prime$: Let $R,S$ ...
6
votes
1answer
281 views
How to show this ideal is not principal
I have been brushing up on cubic number fields. Specifically, let $s$ be a root of the polynomial $x^3 + x^2 + 3x + 17$, and consider $K = \mathbb{Q}(s)$; we have $\mathcal{O}_K = \mathbb{Z}[s]$, and
...
6
votes
1answer
96 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 ...
5
votes
3answers
83 views
Ideal generated by a set in a commutative ring without unity
In a commutative ring with unity $1$, call it $R$, the the ideal generated by the set $S=\{a_1,...,a_n\}$ is the smallest ideal of $R$ containing $S$. It can be proven that this ideal is
$$
...
5
votes
2answers
231 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 ...
5
votes
2answers
125 views
In an ideal, pairwise non-coprime implies globally non-coprime?
Let $R$ be a polynomial ring $R=k[X_1,X_2, \ldots ,X_n]$. Let $I$ be an ideal of $R$ such that any two elements of $I$ have a non-constant gcd. Does it follow that there is a non-constant $D$ dividing ...
5
votes
1answer
143 views
In $K[X,Y]$, is the power of any prime also primary?
I've recently been reading about primary decomposition, and was browsing the questions here.
From this, I know that it is not true that every primary ideal is the power of a prime ideal.
I'm curious ...
5
votes
2answers
152 views
Subset of a P-ideal need not be a P-ideal
I was looking for examples showing that subset of a P-ideal is not necessary. I will post below a counterexample I was able to find. (I hope it is correct.) But I'd be glad to see other simple (or ...
5
votes
1answer
71 views
When is k[x,y]/I complete for the (x,y)-adic topology?
Let $k$ be a field. If necessary, add assumptions on $k$ or just take $k=\mathbb{C}$.
It is easy to classify the ideals $I \subseteq k[x]$ such that $k[x]/I \to k[[x]]/(I)$ is an isomorphism, namely ...
5
votes
2answers
68 views
Question about ideals of rings and rings of n*n matrices
A question that's been bothering me:
R is a ring with unity. Also consider $M_n(R)$ the matrix ring.
If all ideals $J$ of $M_n(R)$ are finitely generated, does every ideal $I$ of $R$ need to be ...
5
votes
2answers
217 views
(Minimal?) Polynomials using the Nullstellensatz
I'm struggling with an exercise that was asked in class:
Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in
the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
5
votes
1answer
65 views
Factorization of $5$ in the splitting field of $x^3 + 2$
I wonder if someone could help to clarify the following.
Let $\zeta$ be a primitive cube root of unity and $\alpha = \sqrt[3]{2}$. Let $K = \mathbb{Q}(\alpha)$ and $L = K(\zeta)$. Then $L$ is the ...
5
votes
1answer
52 views
A prime poset of ideals
Let $A$ be a ring (commutative unital), and $\mathcal I$ be a nonempty family of proper ideals of $A$.
I will say that $\mathcal I$ has property $\dagger$ if for any $\mathfrak a\in\mathcal I$ and ...
5
votes
1answer
147 views
Quotient of ring of integers
Let $R=\mathcal{O}(K)$ be the ring of the integers of $K=\mathbb{Q}[\zeta_8]$, where $\zeta_8=e^{2\pi i/8}=\sqrt{2}/2(1+i)$ is a primitive eighth root of unity in $\mathbb{C}$. It can be shown that ...
5
votes
1answer
104 views
Three maximal ideals lying over $3\mathbb{Z}$?
A few weeks ago I asked a question about finding the number of maximal ideals lying above $3\mathbb{Z}$ in $B$, where $B$ is the integral closure of $\mathbb{Z}$ in a splitting extension ...
5
votes
1answer
150 views
A Gröbner Basis Computation Gone Bad
Here is the problem statement:
Consider the polynomial ideal $I = \langle b-r_1-r_2, c-r_1r_2 \rangle \subset \mathbb{Q}[r_1,r_2,b,c].$ Show that $I \cap \mathbb{Q}[b,c] = \langle 0 \rangle$.
...
5
votes
0answers
137 views
Non-cyclic unit groups
Is there any way to motivate why certain factor rings of $\mathbb{Z}, \mathbb{Z}[i]$, etc., to a prime power have non-cyclic unit groups? For example, the only such non-cyclic unit groups of factor ...
5
votes
0answers
155 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 ...
4
votes
4answers
111 views
Why do $f$ and $f'$ generate all of $K[X]$?
I have been studying Marcus' Number Fields. I am stuck on a remark in Appendix 2, page 258.
He says: A monic irreducible polynomial $f$ of degree n over $K$ (a subfield of $\mathbb{C}$) splits into n ...
