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.
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Composition series of $M_2(\mathbb{R})$ as an $A$-module over itself
Let $A = M_2(\mathbb{R})$ and consider $A$ as the regular $A$-module. Show that $A$ has infinitely many composition series.
The submodules of $A$ as the regular $A$-module are the left ideals of $A$, ...
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Is inclusion of a prime ideal in a non-prime ideal possible?
Let $R$ be a ring which has $p$ as a prime ideal. Can there exist a non-prime proper ideal $m$ of $R$ such that $p$ ⊊ $m$?
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Is $\langle x+y^3,x^3+y \rangle$ a radical ideal in $\mathbb{C}[x,y]$?
Let $u=x+y^3, v=x^3+y$.
Question: Is it true that $I=\langle u,v \rangle$ is a radical ideal?
Of course, $I$ is not a maximal ideal, since its generators have more than one common zero; the common ...
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Pathologies with Product Ideal $I \cdot J $
Let $R$ be an unitary, commutative, Noetherian ring and $I, J$ two ideals.
The product ideal $I \cdot J$ is defined as
$$ I \cdot J:=\{\sum_i^n a_i \cdot b_i \mid a_i\in I,b_i\in J\}$$
note we allow ...
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is the kernel of $\phi(x) = 1 + \sqrt2$ where $\phi : \mathbb{Z}[x] \to \mathbb{R} $ a principal ideal
This is an exercise from Artin chapter 12. I found that the kernel of $\phi$ is principal, as because we are in $\mathbb{Z}[x]$ if $ 1 + \sqrt2$ is a root, so much be $1 - \sqrt2$. Taking $(x-1-\sqrt2)...
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Show that the ideal $\operatorname{rad} (a^2+bc, d^2+bc, (a+d)b, (a+d)c) = (ad-bc,a+d)$ but $(a^2+bc, d^2+bc, (a+d)b, (a+d)c) \not= (ad-bc,a+d)$
We identify the space $M_{2,2}(k)$ of $2\times2$-matrices over $k$ with $\Bbb A^4(k)$ (with coordinates $a,b,c,d$). We define ideals $$\mathfrak{a} := (a^2+bc,d^2+bc,(a+d)b,(a+d)c) \subseteq k[a,b,c,d]...
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Von Neumann regular rings and its ideals
Let R be a strongly regular Von Neumann ring, this is, that for every $r \in R$ there exists $x \in R$ such that $r^2x=r$. From here, how to prove that $R$ is strongly Von Neumann regular if and only ...
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Why the ideal $I=(2,1+\sqrt{-5})$ is not principal with respect to any norm
As you can see in the link, book shows that the ideal $I=(2,1+\sqrt{-5})$ is not principal in $\mathbb Z[\sqrt{-5}]$. I believe that I completely understand the proof however the last sentence makes ...
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Ideal of multivariate polynomial
Let $p(x_1,...,x_n)$ be an element of $\mathbb{F}_2[x_1,...,x_n]/(x_1^2+1,...,x_n^2+1)$. Is there a way to capture the size or dimension of the ideal $(p(x))$? I would guess that if there is a way, it ...
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Commutator Ideal of Toeplitz Operators
Suppose $H^{2}$ is a Hardy space and $T_{\phi}$ is Toeplitz operator on $H^{2}$ with symbol $\phi\in L^{\infty}(\mathbb{T})$. $\mathcal{A}$ is a C*-algebra generated by $\{T_{\phi}:\phi\in L^{\infty}(\...
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What is the norm of an ideal $(2,1+\sqrt{-5})$ in $\mathbb Z[\sqrt{-5}]$?
Question: What is the norm of an ideal $I=(2,1+\sqrt{-5})$ in $\mathbb Z[\sqrt{-5}]$?
Basically, I use the usual norm. $N(a)=a.\overline a=a^2+5db^2$
I know that if ideal is generated by one element ...
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What ring homomorphisms do induce well-defined mapping between principal ideals?
The original question I was going to ask was an attempt to define the notion of a "GCD domain homomorphism", but ultimately it boiled down to another question, so here I go.
Given a ring $R$,...
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Proof that ring of rational upper triangular matrices is a right Goldie ring
Let $R = T_n (\mathbb{Q})$ - ring of rational upper triangular matrices. I want proof that it's a right Goldie ring, i.e. it satisfies two conditions:
Any chain of right annihilators is stabilized: $\...
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Prove that $V(I(X)) = X$
Let $A$ be a ring and $X \subset SpecA$ a closed set.
Prove that $I(X) := \bigcap P \in XP$ is a radical ideal.
My proof is as follows:
$I(X)$ is an ideal since it's the intersection of ideals. If $f^...
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I am getting very confused by the definition of a minimal prime ideal
I am trying to prove that in an affine scheme $\operatorname{Spec}A$ that an irreducible component can be written as the vanishing locus $V(\mathfrak p)$ for a minimal prime ideal $\mathfrak p$, but I ...
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