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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1answer
24 views
Question regarding nilpotent ideals of a ring.
I am working on the following:
An ideal $N$ is called nilpotent if $N^n$ is the zero ideal for some $n\geq1$. Prove that the ideal $p\mathbb{Z}/p^m\mathbb{Z}$ is a nilpotent ideal in the ring ...
3
votes
1answer
76 views
Norm of ideals in quadratic number fields
I do not really understand how to factor ideals in a quadratic field $K = \mathbb{Q}(\sqrt{d})$, mainly because I have some trouble computing the norm of ideals. I think I understand what is going on ...
3
votes
1answer
80 views
Gröbner Basis for Ideal $J$
I have the following question...
Consider the ideal $J:= (x^2y-x^2y^2,\ x^2z-z^2yx,\ x^2+xz) \subset \mathbb{Q}[x,y,z]$
Is $x \in J?$ Is $x \in \sqrt{J} $?
I know finding if $x$ is in the radical ...
3
votes
2answers
39 views
Ideals with two inputs
What does it mean if an Ideal $I$ is generated by two inputs. Like, let $I = (p, x^2 +1)$. And then, for a Ring $R$, what does $R/I$ mean?
3
votes
2answers
53 views
Proof of $R/I$ integral over $S/(S \cap I)$
Can you tell me if my reasoning is correct?
I want to prove if $S \subset R$ are rings and $R$ is integral over $S$ and $I$ is an ideal of $R$ then $R/I$ is integral over $S/ (S\cap I)$.
Let $R$ be ...
3
votes
1answer
330 views
When does a ring homomorphism preserve ideals
In my textbook they say that if $f: R \rightarrow S$ is a ring homomorphism, then:
if $I \subset R$ is an left ideal of $R$, then $f(I)$ is a left ideal of $S$.
However, I think that this is a ...
3
votes
1answer
46 views
Proving a factorization of ideals in a Dedekind Domain
Let $R=\mathbb{Z}[\sqrt{-13}]$. Let $p$ be a prime integer, $p\neq 2,13$ and suppose that $p$ divides an integer of the form $a^2+13b^2$, where $a$ and $b$ are in $\mathbb{Z}$ and are coprime. Let ...
3
votes
1answer
59 views
Radical/Prime/Maximal ideals under quotient maps
Let $I$ be an ideal of a ring (commutative with unity) $R$ and let $q:R\to R/I$ be the quotient map. Then there is a well known correspondence between ideals of $R$ containing $I$ and ideals of $R/I.$ ...
3
votes
2answers
143 views
Points and maximal ideals in polynomial rings
Let $k$ be a field, then I want to prove the following statement: for every $P=(b_1,\ldots,b_n)\in K^n$, the ideal $\mathfrak{m}_P=(x_1-b_1,\ldots,x_n-b_n)$ is maximal in the polynomial ring ...
3
votes
1answer
55 views
Cyclic representation
Suppose $V\neq0$ is a representation of an algebra A.
Definition: $v\in V$ is cyclic if and only if it generates $V$, thus $Av=V$. If a representation has a cyclic vector we call the representation ...
3
votes
2answers
70 views
How to compute dimension of a space?
Let $I\subset S=K[X_1,X_2,\dots,X_n]$ be a monomial ideal.
(a) Show that $\dim_K S/I < ∞ ⇔ ∃a∈\mathbb{Z}_+ : X^a_i \in I ∀i$.
(b) Given integers $a_i\in \mathbb{Z}_+$, compute $\dim_K S/I$ for ...
3
votes
1answer
201 views
Diophantine equation (use class ideal group to solve)
Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$
My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
3
votes
1answer
120 views
Why is $S/Z$ a domain for the ideal $Z=\{z\in S\mid za=0,\;\forall a\in R\}$ in $S$?
Suppose $R$ is a rng with no zero divisors, not necessarily commutative. I know $R$ can be embedded into a ring $S:=\mathbb{Z}\times R$ by identifying $r\in R$ with $(0,r)\in S$. The operations on $S$ ...
3
votes
1answer
133 views
Generating set for sum of two ideals
Suppose there are two ideals $I,J \in \mathbb{C}[x_1,\dots,x_k]$ and two sets of generating polynomials $\langle f_1, \dots, f_s\rangle$, $\langle g_1, \dots, g_t\rangle$. Now I want to describe $I + ...
3
votes
1answer
118 views
Minimal generating sets for homogeneous polynomial ideal in two variables
This question is (somehow) related to System of generator of a homogenous ideal
Let $A$ be the ring $A={\mathbb R}[X,Y]$, and let $m \geq 1$. Let
$$
{\cal S}_m=\lbrace X^m, X^{m-1}Y,X^{m-2}Y^2, ...
3
votes
2answers
79 views
Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$
I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$.
I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
3
votes
1answer
54 views
Computing kernel of ring homomorphism
I am trying to answer the question already asked here.
My question is two parts:
1) I think I have found a proof on my own, could someone check it is valid?
Modulo that ideal, $x_i\equiv a_i$ so ...
3
votes
1answer
111 views
Proof about noetherian rings
I have to prove that every finite ring is noetherian. I know examples of noetherian rings which are not finite such as the field of complex numbers or a PIR like the integers. But anyway:
[Proof]:
I ...
3
votes
1answer
44 views
Radical Ideals: Show that $\sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I+J}$
The direction $\sqrt{\sqrt{I}+\sqrt{J}}\supset \sqrt{I+J}$ is trivial as $\sqrt{I}+\sqrt{J} \supset I+J$ since $\sqrt{K} \supset K$ for any ideal $K$. Is the following correct?
My attempt: Suppose ...
3
votes
1answer
87 views
Classifying all ideals of a lattice $\mathbb{Z}[\sqrt{-d}]$
In Artin's Algebra he presents a method (that I am sure I am butchering) for classifying ideals of a given lattice $\mathbb{Z}[\sqrt{-d}]$ by taking any ideal $I$, choosing an element of minimum norm ...
3
votes
1answer
70 views
Equivalence of Valuations - Trouble Understanding Proof
I want to complete the proof of the following theorem. Here is what I have got so far:
Theorem Every non-euclidean valuation $v$ on a number field $K$ is equivalent to $v_{\mathfrak p}$ for some ...
3
votes
0answers
35 views
Finiteness of ideal of given norm
I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$.
I know there are "standard proofs" (eg How many elements ...
3
votes
0answers
51 views
primary decomposition of ideals
How to find the primary decomposition of ideal $I = (X^2, XY, XZ, YZ)$ in the ring $k[X,Y,Z]$?
Also how to show that $(X,Y)^{308}$ is primary ideal in $k[X,Y,Z]$? Is there a general rule for finding ...
3
votes
0answers
48 views
Proof about affine varieties
Ok so I have that $k$ is algebraically closed and $F$ is an element of $k^n$, and it is a reduced polynomial. We have that $V = V(F)$. In the book it says prove that $F$ generates $I(V)$ but in my ...
3
votes
0answers
59 views
In relation with the set of polynomially Fredholm perturbation elements
Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of polynomially Fredholm perturbation elements in $A$, i.e.
...
3
votes
0answers
20 views
Is there an ideal decomposition that counts the number of monomial generators?
Consider the ideal $I\subseteq S[x,y,z]$ where $S$ is some field of characteristic 0 (probably any field will do) and $I=<x^9-y^4z^4,y^9-x^5z^4,z^8-x^4y^5,x^6>$. Notice that because the lone ...
3
votes
0answers
119 views
In a PID without unit an ideal is maximal iff it is prime.
As the titles says, I need to show that in a PID $R$ an ideal is maximal iff it is prime. This is easy to do if $R$ has a multiplicative identity. I can not do it if $R$ does not have an identity. It ...
3
votes
0answers
107 views
When is intersection of infinitely many maximal ideals zero?
I've been trying without success to figure out what are the rings $R$ such that whenever $M_n, n \in \omega$ is a countably infinite collection of pairwise distinct maximal ideals then $\bigcap_{n \in ...
3
votes
1answer
167 views
admissible ideals
How to prove the following conclusion :
For any finite quiver $Q$, an ideal $I$ of $KQ$, contained
in $R^2_Q$, is admissible if and only if, for each cycle
$\sigma$ in $Q$, there exists $s \geq 1$ ...
3
votes
0answers
121 views
Question about proof of Going-down theorem and prop. 3.16 in AM
Prop. 3.16 tells us that if $f: A \to B$ is a ring homomorphism and $\mathfrak p$ is a prime ideal of $A$ then $\mathfrak p$ is the contraction of a prime ideal of $B$ if and only if $\mathfrak p^{ec} ...
3
votes
0answers
106 views
Can the ideal $(X_1, X_2, \dots, X_n) $ be generated by fewer polynomials over the field $K[X_1, X_2, \dots, X_n]$?
My algebra teacher asked whether the ideal $(X_1, X_2, \dots, X_n) $ can be generated by fewer polynomials over the field $K[X_1, X_2, \dots, X_n]$.
My intuition tells me that it can't, so I tried to ...
2
votes
3answers
103 views
What's the motivation of the ideal? [duplicate]
I'm reading a book on Algebra, it introduces the concept of ring after some examples, the concept of ideal.
Definition I.1.8. Let $(A,+,\cdot)$ be a ring and $I$ a non-empty subset of $A$. We say ...
2
votes
5answers
355 views
The image of an ideal under a homomorphism may not be an ideal
This is an elementary question about ideals. Consider a ring homomorphism
$$
f: \mathbb{Z} \rightarrow \mathbb{Z}[x],
$$
and consider the ideal $\left< 2\right>$ in $\mathbb{Z}$. When why is ...
2
votes
2answers
97 views
This ideal is prime. But is it also maximal?
Could someone please give me a hint how to prove (preferably directly, without finding clever homomorphisms) that $(X,Y)$ is a maximal ideal in $\mathbb{C}[X,Y]$ ?
This ideal is prime, since it ...
2
votes
3answers
67 views
Affine variety over a field
Suppose we have an algebraically closed field $K$. An affine variety is the common zero locus of a collection of polynomials $f_{\alpha} \in K[z_1, \dots, z_n]$. So basically it is the set of points ...
2
votes
3answers
106 views
Proper ideals in $\mathbb{Q}[x,y,z]$
I am trying to show that the ideal $I = (x^2 -2, y^2 +1, z)$ is a proper ideal of $\mathbb{Q}[x,y,z]$. I have been trying to show that 1 is not in the ideal with a degree argument and evaluating at a ...
2
votes
3answers
453 views
Ideal of the twisted cubic
The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute ...
2
votes
2answers
593 views
3 different prime ideals in Z[x]
find 3 different prime ideals in $Z[x]$, $I,J,K$ such that $I\subset J\subset K$.
have no clue where to start from.
2
votes
2answers
182 views
What does the word “ideal” mean in this context?
I'm confused about a terminology.
In Frank W. Warner's book Foundations of Differentiable Manifolds and Lie Groups, it says on page 12
Let $F_m$, a subset of $\bar{F_m}$ (the set of germs at ...
2
votes
1answer
327 views
When is a product of two ideals strictly included in their intersection?
Let $I,J$ two ideals in a ring $R$. The product of ideals $IJ$ is included in $I \cap J$. For example we have equality in $\mathbb{Z}$ if generators have no common nontrival factors, in a ring $R$ ...
2
votes
2answers
83 views
Can one define “addition” of ideals to correspond to addition of numbers?
(In the setting of number fields and algebraic integers) If $(a),(b)$ are two principle ideals then $(a)+(b)$ corresponds to $(\gcd(a,b))$, so while the natural definition of addition for ideals has a ...
2
votes
2answers
91 views
Product of ideals corresponding to vanishing of points is equal to their intersection
Let $k$ be some field, and let $v,v',v''$ be three distinct points in $k^3$. Let $\mathfrak{m}_v = (X_1 - v_1,X_2 - v_2,X_3 - v_3)$ be the ideal in $k[X_1,X_2,X_3]$ corresponding to the polynomials ...
2
votes
2answers
66 views
$ I(J+L)=IJ+IL$ if $I,J,L$ are ideals of $K$
Given that $I,J,L$ are ideals of $K$, do we have $I(J+L)=IJ+IL$?
I am confused how to do it.
2
votes
1answer
198 views
Find all the prime ideals of $\{\frac{a}{b}| a \in \mathbb{Z}, b \in \mathbb{N}_0 \text{ odd}\}$
For an exercise in my book I have to find all the prime ideals of $$R = \left.\left\{\frac{a}{b}\;\right|\; a \in \mathbb{Z}, b \in \mathbb{N}_0 \text{ odd}\right\}\leq (\mathbb{Q},+,\cdot)$$
I ...
2
votes
1answer
189 views
Ideal in the product of two rings:
$R$ and $S$
are two ring, let $J$
ideal in $R\times S$
then there are $I_{1}$
ideal of $R$
and $I_{2}$
ideal of $S$
such that $J=I_{1}\times I_{2}$
For me is abvious why $\left\{ r\in ...
2
votes
2answers
313 views
Finding a primary decomposition
Let $k$ be a field, and define $R=k[x,y]$.
I'm supposed to find two different **minimal **primary decompositions of the ideal $(x^2y, y^2x)$.
It's easy to see that one minimal primary decomposition ...
2
votes
3answers
144 views
Showing an ideal is a projective module via a split exact sequence
Let $R=\mathbb{Z}[\sqrt{-6}]$ and $I=(2,\sqrt{-6})$ the ideal generated by $2$ and $\sqrt{-6}$.
I want to show that $I$ is a projective $R$-module by producing a short exact sequence that splits, ...
2
votes
2answers
100 views
Let $R$ be a Dedekind domain. If $I$,$J$ are $R$-ideals, prove that there exists an $\alpha\in I$ such that $\gcd(\langle\alpha\rangle,IJ)=I$
Let $R$ be a Dedekind domain. If $I$,$J$ are $R$-ideals, prove that there exists an $\alpha\in I$ such that $\gcd(\langle\alpha\rangle,IJ)=I$.
Well if I could prove that there existed an $\alpha\in ...
2
votes
1answer
56 views
For which $m \in \mathbb N$ is the ideal $(m,x^2+y^2)$ prime in $\mathbb Z[x,y]$?
Let $m \in \mathbb N$. Find a necessary and sufficient condition for
$m$ such that the ideal $(m,x^2+y^2)$ is prime in $\mathbb Z[x,y]$.
I have to find for which $m$ the quotient ring is an ...
2
votes
2answers
45 views
$(ax)(ay) = a(xy) \in (a)$
Lemma 26.1
Let $R$ be a commutative ring with unity element $e$. The set
$(a) = \{ar : r \in R\}$
is an ideal of $R$.
Proof.
First, we will show that $(a)$ is a subring of $R.$ Since $a = ae$ then $a ...
