All Questions
Tagged with ideals linear-algebra
85
questions
2
votes
2
answers
47
views
What is the semidirect product we use in Levi Decomposition
So using the Levi Decomposition for any lie algebra $\mathfrak{g}$, there exists a semisimple subalgebra $\mathfrak{s}$ such that:
$Rad(\mathfrak{g})$$\ltimes$$\mathfrak{s}$=$\mathfrak{g}$
However in ...
- 69
2
votes
1
answer
50
views
Ideals of the upper-triangular matrix algebra $\mathcal{T}_n$ over $\Bbb{R}$ or $\Bbb{C}$
Let $\Bbb{F}$ be the field of all real or complex numbers and let $\mathcal{T}_n$ be the algebra of all upper-triangular $n\times n$ matrices over $\Bbb{F}$. Can one completely describe all (two-sided)...
- 46.7k
0
votes
0
answers
55
views
An injection from the set of normal subgroups to subsets of irreducible representations
Some things I understand to be true:
(1) A finite dimensional $\mathrm{C}^*$-algebra $A$ is of the form
$$A\cong \bigoplus_{j=1}^NM_{n_j}(\mathbb{C})\qquad (n_j\in \mathbb{N}).$$
(2) With respect to ...
- 7,789
-1
votes
1
answer
59
views
Find all ideals of $\mathbb{Z}[\frac{1}{2}]$ [duplicate]
The ring $\mathbb{Z}[\frac{1}{2}]=\{\frac{u}{2^n}\}$ is the localization of $\mathbb{Z}$ in regards to $S=\{2^n:n \in \mathbb{N}\}$. Find all ideals of this ring.
I'm not sure what is meant by this ...
- 65
0
votes
0
answers
17
views
Set of co-images form left ideal of ring
Let $K$ be a field and $R=K^{2 \times 2}$.
Let $U$ be a subspace of $\mathbb{Q}^2$ and $L=\{A \in R: coim(A) \subset U\}$. Then $L$ is a left ideal of $R$.
For $L$ to be a left ideal the difference of ...
- 65
0
votes
1
answer
59
views
Existence of ideal
Let $R$ be a ring, $I$ a finitely generated ideal of $R$ and $J$ an ideal of $R$ with $J \subset I$. Show that $R$ has an ideal $M$ so that $J \subset M \subset I$ and for every ideal $N$ with $M \...
- 65
0
votes
0
answers
35
views
Let V be a finite-dimensional vector space, $End_{K}(V)$ the ring of endomorphisms. [duplicate]
Let $V$ be a finite-dimensional vector space, $End_{K}(V)$ the ring of endomorphisms. Furthermore, let $I$ be an ideal of $End_{K}(V)$, $I \neq {0}$.
I want to show that for all $u, v \in V, u,v \neq ...
- 351
0
votes
1
answer
119
views
Is the ring $\mathbb{R} [x]$ Noetherian?
I aim to assess the soundness of my approach. In an exercise, the question is posed regarding whether $\mathbb{R}[x]$ is Noetherian, and my response is negative. To support this, I employ the ...
- 123
3
votes
2
answers
114
views
What is (if any) the generalization of the concept of a basis for ideals of rings?
This question is motivated by the following problem from a problem set on ring theory:
Prove that, if $R$ is a Noetherian commutative ring, and $I$ is its ideal, $R/I$ is a again Noetherian.
My ...
- 1,197
0
votes
0
answers
101
views
Let $R$ be a factorial ring in which every ideal generated by two elements is a principal ideal. Show that $R$ is a principal ideal ring. [duplicate]
I want to check if my solutions for this problem are right.
Let $R$ be a factorial ring in which every ideal generated by two elements is a principal ideal.
Show that $R$ is a principal ideal ring.
...
- 900
1
vote
1
answer
66
views
Is there a proof for the existence of a complete flag of solvable Lie algebras independet on lie's theorem?
Basically title. Is there proof for the following statement, which doesn't rely on Lie's-Theorem?
For any solvable Lie-Algebra L, there exists a Flag of Ideals $0=I_0\subset \dots\subset I_n=L$ with $\...
- 364
0
votes
2
answers
81
views
a doubt on the left ideal of the matrix ring in Dummit&Foote's Abstract Algebra
I have a doubt in Dummit's Abstract Algebra on page245 :
"For each $j\in[1,2,\dots,n]$ let $L_j$ be the set of all $n\times n$ matrices in $M_n(R)$ with arbitrary entries in the $j^{\text{th}}$ ...
- 361
0
votes
1
answer
29
views
Show statements about "sums" and "square roots" of ideals
Let $R$ be a commutative ring with $1$ and $I,J$ ideals of $R$. Show:
(1) $I+J:=\{i+j|i \in I,j \in J\}=\langle I \cup J \rangle_R$
(2) $\sqrt{I} := \{x \in R | x^n \in I, n\in \mathbb{N}\}$ is an ...
1
vote
1
answer
39
views
Check if the ideal generated by the set is equal to the ring
Let $\langle S \rangle_R$ be the ideal of $R$ generated by the set $S$.
$1.R=\mathbb{Z},S=\{105,70,42,30\}$
$2.R=\mathbb{Z} \times \mathbb{Z},S=\{(4,3),(6,5)\}$
$3.R=\mathbb{Z}_{101}, S=\{[75]_{\equiv ...
0
votes
1
answer
325
views
Find all left ideals of $M_n(\mathbb{C})$
I am taking an introductory abstract algebra class, closely following Gregory T. Lee's Abstract
Algebra // An Introductory Course.
I have been asked to find all left, right, and two-sided ideals of ...
- 978