Questions tagged [nilpotence]

A nilpotent element of a ring has $a^n=0$ for some integer $n$.

Filter by
Sorted by
Tagged with
3 votes
1 answer
54 views

Which Lie algebras are derived algebras?

Let $\mathfrak{h}$ be a Lie algebra over a zero characteristic algebraically closed field. When is $\mathfrak{h}$ the derived algebra of some Lie algebra $\mathfrak{g}$? This clearly imposes some ...
  • 4,186
0 votes
1 answer
71 views

Subspace of square matrices where all matrices are nilpotent and the products of the matrices are also in the subspace.

Let $A$ and $B$ be $3\times3$ square matrices (maybe could also examine $n\times n$) and let $U$ be a subspace where $A,B,AB\in U$. All matrices in $U$ are nilpotent. I conjecture that all matrices in ...
  • 183
0 votes
0 answers
22 views

Proving if $I$ nilpotent and $I\neq N(R)$ then $R/I$ has nilpotents

I am trying to prove that if $R$ is a ring and $I$ is a nilpotent ideal of $R$ and $I\neq N(R)$ then $R/I$ has a nonzero nilpotent element. My attempt Since $I$ nilpotent ideal of $R$ we have $I \...
  • 1,824
0 votes
0 answers
45 views

Computing the nilradical of a ring

Let $R=\begin{pmatrix} \mathbb{C} & \mathbb{C} & \mathbb{C}\\ 0 & \mathbb{C} & \mathbb{C}\\ 0 & \mathbb{C} & \mathbb{C} \end{pmatrix}$. I want to find the nilradical: $$N(R)=\...
  • 115
1 vote
0 answers
62 views

Square root from nilpotents

We know that $$\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)^2=\left( \begin{...
  • 8,320
3 votes
0 answers
54 views

Fulton & Harris Proof Clarification for Lemma D.6

Lemma D.6 of Fulton & Harris claims If $H$ is regular, then $\mathfrak g_0(H)$ is abelian, where $\mathfrak g$ is a complex Lie algebra, $\mathfrak g_0(H)=\ker\operatorname{ad}(H)^m$ for large $...
  • 6,024
2 votes
0 answers
50 views

Matrix multiplication and nilpotent matrix

Given that you have an $n\times n$-matrix called $A$ with the property that $A^k = 0$. Prove that $A^n = 0$. Have heard about eigenvalues and null space as ways for the proof. The thing is that I can'...
4 votes
1 answer
72 views

For which $n \in \mathbb N$ does $A = \mathbb Z/ n \mathbb Z$ has $\operatorname{Nil}(A) \neq 0$?

I want to try to answer the following question: For which $n \in \mathbb N$ does $A = \mathbb Z/ n \mathbb Z$ has $\operatorname{Nil}(A) \neq 0$? Here are my thoughts: I know that $\operatorname{Nil}(...
0 votes
0 answers
38 views

Is there a name for the sum of a scalar + nilpotent operator?

Let $V$ be a vector space and suppose $T\in\operatorname{End}(V)$ is the sum of a scalar operator and a nilpotent operator. That is, $T=T_s+T_n$, where $T_s=\lambda I$ for some scalar $\lambda$, and $...
  • 6,024
0 votes
1 answer
65 views

Nilpotent elements in $\mathbb{R}[x]/\langle x^4+4\rangle$

Let $\mathbb{R}[x]$ denote the ring of polynomials in $X$ with real coefficients. Then, the quotient ring $R=\mathbb{R}[x]/(x^4+4)$ is (a) field (b) an integral domain, but not a field (c) not an ...
2 votes
2 answers
32 views

Nilpotent Matrix And Sequence Properties

The original problem from algebra book asks to prove that if $A$ is a $2\times2$ nilpotent matix then $A^2=0$ . Can this be related with some properties of exact sequences ? I.m. that such matrix is a ...
0 votes
1 answer
31 views

Is it ok to have a nilpotent matrix with index $0$, and is there a short-cut to find the index?

I am thinking about the definition of a nilpotent matrix, what I know is that a nilpotent matrix is a square matrix, say matrix $A$, such that $A^k$ is a null matrix with index $k$. According to this, ...
1 vote
1 answer
61 views

How to find in practise a $\mathfrak{sl}_2$-triple in a simple lie algebra?

I want to justify the existence of a $\mathfrak{sl}_2$-triple in a simple lie algebra. I know there exists the Jacobson-Morozov theorem. It states that given a nilpotent element of the lie algebra, ...
  • 67
2 votes
1 answer
27 views

Nilradical in an Artinian ring

I am reading a book that suggests to find what $N(R)$ is - the nilradical of $R$ where $R$ is Artinian you just need to find a nilpotent ideal $I $ of $R$ and then show that $R/I $ has no nonzero ...
  • 1,824
1 vote
1 answer
30 views

Nilpotent ideals in a ring proof

I am trying to understand a proof I am reading but it doesn’t make much sense to me. If $R$ is a ring with $I$ a nilpotent left ideal then $I$ is contained in a nilpotent two-sided ideal. The proof ...
  • 1,824
1 vote
1 answer
28 views

Is $N_T \subseteq Im(T^l)$?

If $T\in \mathcal L(V)$ is a nilpotent linear map (ie. $T^k=0_V$ for some $k \in \mathbb{N}_{\ne 0}$), let $N_T$ be the nullspace of $T$, $Im(T^l)$ be the image of $V$ under $T^l$. Question: Is $N_T \...
  • 139
2 votes
0 answers
119 views

When is a family of nilpotent matrices over $\mathbb{F}_p$ zero?

Given some $n\times n$ matrices $ A_{i},i=1,\dots,m $, over a finite field $\mathbb{F}_p$. Suppose that all $ A_{i} $ are nilpotent and $$ c_{i}A_{i}+\sum_{j\neq i}b_{ij}[A_{i},A_{j}]=0 $$ for some $ ...
1 vote
0 answers
56 views

show that $AB-BA$ is nilpotent [duplicate]

let the set of all n by n matrices with complex entries be denoted by $M_n(\mathbb{C}).$ Let $A,B\in M_n(\mathbb{C})$ be such that $AB-BA $ commutes with $A$. Show that $AB-BA$ is nilpotent. I need ...
  • 1,689
-1 votes
1 answer
51 views

Ideals generated by all nilpotent elements in noncommutative rings

It is a basic fact that a set of all nilpotent elements in a commutative ring is an ideal. Suppose that $A$ is a noncommutative ring, $I$ is a two-sided ideal generated by all nilpotent elements and $...
1 vote
1 answer
39 views

Let $R=\mathbb Z_{36}$. Find $N(\langle 0 \rangle), N(\langle 4\rangle), N(\langle 6 \rangle)$.

Let $R=\mathbb Z_{36}$. Find $N(\langle 0 \rangle), N(\langle 4\rangle), N(\langle 6 \rangle)$. Let $R$ be a commutative ring and let $A$ be any ideal of $R$. Then the nilradical of A, $N(A)=\{r\in R:...
2 votes
3 answers
340 views

Invertible element can't be nilpotent?

When I was reviewing cracking GRE subject mathematics 4-th edition, I was confused about the proof in Page 247. Consider invertible element c in a Ring, $cc^{-1} = 1$. Then for any integer n, $(cc^{-...
3 votes
1 answer
76 views

Fitting subgroup of any group is locally nilpotent

Is it true that the Fitting subgroup of any group is locally nilpotent? I know that if $G$ is finite then $F(G)$ is nilpotent and so locally nilpotent. Also, if $G$ is infinite then there's exists ...
  • 559
1 vote
0 answers
20 views

Signature of endomorphism [closed]

I need some help on how to do this question please: Let N ∈ M$_n$(F) be a nilpotent matrix and consider the endomorphism T induced on M$_n$(F) by the rule: A → NA. How is the signature of T related to ...
  • 35
3 votes
5 answers
147 views

$A$ is an $n$ by $n$ matrix over $\mathbb C$ such that $Rank(A)=1$ and $Tr(A)=0$, prove that $A^2=0$

This is what I did: Because $\mbox{rank}(A)=1$, then from rank nullity theorem $$\dim\ker A + \mbox{rank} A = n \implies \dim \ker A = n-1$$ and $gm(0)=dimE(0)=dimKerA=n-1$ where $E(0)$ is the ...
7 votes
2 answers
119 views

Is every non-diagonalizable matrix diagonalizable over a larger non-reduced commutative ring?

Let $A$ be a square matrix with entries in some algebraically closed field $K$ (e.g., the complex numbers $\mathbb{C}$) and suppose that $A$ is not diagonalizable (so there is at least one eigenvalue ...
2 votes
1 answer
28 views

Existence of maximal ideal in a commutative ring

Let $A$ be a commutative ring, $I \subsetneq A$ a proper ideal of $A$ and $a \in A$ such that $a^k \neq 0$ for all integer $k > 0$. Then there exists an ideal $J$ of $A$ that is maximal satisfying $...
  • 1,422
3 votes
1 answer
37 views

Rank of limit points of a conjugacy class

Problem: Let $t \to P_t$ be a one parameter subgroup $\mathbb{C}^* \to \text{Gl}_{n}(\mathbb{C})$. Let $X$ be a $n \times n$ nilpotent matrix. I want to show that if $lim_{t \to 0} P_t^{-1}XP_t = Y$ ...
0 votes
0 answers
59 views

Prove an endomorphism is nilpotent [duplicate]

Let $V$ a finite dimensional vector space over a characteristic zero algebraically closed field. Let $x,y\in\mathfrak{gl}(V)$ and $[x,y]=z$ such that $z$ commutes with $x$ and $y$. Prove $z$ is a ...
  • 103
0 votes
2 answers
85 views

Is it true that nilpotent always has some eigenvalue?

I understand that if a nilpotent matrix has some $\lambda$ eigenvector, then it implies that $\lambda=0$ because if $$Ax = \lambda x \\ A^2x = \lambda^2x \\ A^3x=\lambda^3x \\ \vdots \\ 0=A^k=\lambda^...
  • 31
2 votes
1 answer
59 views

If $A$ has no non-trivial idempotents, then neither does $A/N$

Let $A$ be a commutative, associative, unital, finitely generated algebra over an algebraically closed field $k$. Denote by $N$ the nilradical of $A$, which is the set of all nilpotent elements of $A$ ...
  • 2,308
1 vote
1 answer
31 views

Unipotent and semisimple elements are locally finite

Let $k$ be an arbitrary field, and let $V$ be an arbitrary $k$-vector space, possibly infinite-dimensional. Let $g\in\operatorname{End}_k(V)$. Then: $g$ is diagonalizable if $V$ has a basis of ...
  • 2,308
0 votes
1 answer
46 views

Constructing pairwise commuting nilpotent matrices

How can I construct $K$ mutually commuting nilpotent matrices $A_i$ with nilpotent index 3? In other words, I need a set of matrices $A_i$ with following properties: $A_i^3=0$ for $1\leq i \leq K$ $...
1 vote
0 answers
99 views

Does $AX+XB=B$ have a unique solution?

Let $A, B \in \Bbb C^{n \times n}$ such that $A$ is nilpotent but $B$ is non-nilpotent satisfying $AB+BA=O$. Then, $AX+XB=B$ has unique solution. True or false? I don't understand how to approach ...
0 votes
1 answer
47 views

Kernel and image of a nilpotent linear map

My question is as follows. Let $T : V \to V$ be a linear map of a finite dimensional vector space V. If $ker T = Im T$, then is $dim V$ even and $T^{2} = 0$? I believe this to be false since I have ...
0 votes
0 answers
41 views

What is the index of nilpotency of $A$?

Here is the question I am trying to answer: Find the Jordan form of $$A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 0\\ 0 & 0 & -1 \end{pmatrix}$$ I know that the first step is to ...
  • 1,351
0 votes
1 answer
149 views

Prove if $T$ is nilpotent, $ST=TS$, and $\dim\operatorname{Nul}T=2$, then $S$ has at most $2$ distinct eigenvalues

Let $V$ be a finite-dimensional vector space, and let $S,T \in L(V)$ which commute, i.e. $ST = TS$. Suppose $T$ is nilpotent with $\dim\operatorname{Nul}T = 2$. Prove that $S$ has at most $2$ distinct ...
  • 1,103
0 votes
0 answers
49 views

Constructing a strictly upper triangular nilpotent matrix

I asked previously about proving a lemma, where one can choose a basis for a linear operator such its matrix representation in this basis is strictly upper triangular. My original post is here: How to ...
  • 8,397
0 votes
0 answers
51 views

How to show that for nilpotent linear operator $T$, there is a basis with respect to which the matrix of $T$ is strictly upper triangular [duplicate]

I need to prove the following: Lemma: Suppose $T$ is nilpotent linear operator on $n$-dimensional complex vector space $V$. Then there is a basis of $V$ with respect to which the matrix of $T$ ...
  • 8,397
0 votes
1 answer
410 views

if $ \dim \ker T^{k-1} < n , \dim \ker T^k = n $ and $ \dim V=n $ then $T$ is nilpotent

Statement: Let $ T: V \to V $ be a linear transformation over a finitely generated vector space $ V $. Suppose there exists $ k \leq n $ s.t. $\dim \ker T^{k-1} < n , \dim \ker T^k = n $ and $ \dim ...
  • 1,625
2 votes
2 answers
91 views

Is it possible for $AA^T$ to be a nilpotent matrix if neither $A$ nor $A^T$ are?

If $A$ is a square, non-nilpotent matrix with real-valued elements, and its transpose is $A^T$, then is it ever possible for $AA^T$ to be nilpotent? What if we allow complex-valued elements? Is $AA^H$ ...
  • 1,277
1 vote
3 answers
164 views

Construct a $2022\times 2022$ matrix $A$ such that $A^{2022}=0$ but $A^{2021} \neq 0$

I want to construct a $2022\times 2022$ matrix $A$ such that $A^{2022}=0$ but $A^{2021} \neq 0$. More generally, I want to know how to construct a $n\times n$ matrix $A$ such that $A^n=0$ but $A^{n-1}...
  • 6,134
2 votes
3 answers
127 views

If a Lie algebra $\mathfrak{g}$ with only one Cartan subalgebra, then can we implies that $\mathfrak{g}$ is nilpotent?

We suppose that $\mathfrak{g}$ is a finite dimensional Lie algebra over $\mathbb{C}$. My professor told me that the assertion was clear, but I cannot figure it out. I know that each two Cartan ...
0 votes
1 answer
45 views

$T \in \text{Hom}V $ is nilpotent implies $I - T$ invertible [duplicate]

Assume $T \in \text{Hom}V$ is nilpotent I.e. $T^n=0$ for some $n$. How do you show that this implies $I - T$ is invertible? The hint says the proof has something to do with the power series but I’m ...
  • 133
0 votes
1 answer
61 views

If $N^n=0$ but $N^{n-1}\neq 0$, then there is no $n\times n$ matrix $A$ such that $A^2=N$

Let $n\in \mathbb N$ and let $N$ be an $n\times n$ matrix over the field $F$ such that $N^n=0$ but $N^{n-1}\neq 0$. Prove that there is no $n\times n$ matrix $A$ such that $A^2=N$. I can understand ...
  • 6,859
0 votes
0 answers
35 views

When does superdiagonal matrix have a single Jordan block

Let $A$ be the superdiagonal matrix given by $A=(a_{ij})$ where $a_{i,i+1}=r_i$ are nonzero entries and all the other entries are $0$. When is $A$ similar to a single Jordan block with eigenvalue $0$ ...
1 vote
2 answers
106 views

Clarification regarding the number of Jordan blocks of some specific order

Good day! Last week at my algebra class I was asked to prove (and find) the following: Given some nilpotent operator $T$ prove that for some number $k$, the number of Jordan normal blocks of order at ...
0 votes
1 answer
71 views

Linear algebra: nilpotent matrix and determinant.

My linear book had an exercise that demonstrated that a nilpotent matrix A has det(A)=0 $A^k=0$ is the nilpotent condition. $det(A^k)=(det(A))^k$ and since $det(A^k)=0 \Leftrightarrow det(A)=0$. My ...
1 vote
1 answer
85 views

$ N$ is nilpotent and $ T $ is diagonalizable transformation s.t. $ N \circ T = T \circ N $. Show $ \ker q_{\lambda} ( T + N ) = V_{\lambda} $

Problem: Let $ V $ be finitely generated vector space over $ \mathbb{F} $, let $ T: V \to V $ be a diagonalizable linear transformation and $ N : V \to V $ a nilpotent linear transformation and ...
  • 1,625
1 vote
1 answer
103 views

Let $ T: V \to V $ be nilpotent. Let $ f \in \mathbb{F[x]}. $ Show that $ f(T) $ is invertible iff $ f(0) \neq 0 $

Theorem: Let $ T: V \to V $ be nilpotent. Let $ f \in \mathbb{F[x]}. $ Show that $ f(T) $ is invertible iff $ f(0) \neq 0 $ Attempt: $ ( \leftarrow ) $ Suppose that $ f(T) $ is invertible. Hence ...
  • 1,625
0 votes
0 answers
52 views

$R[x]$ is semiprimitive if and only if $R$ has no non-zero nilpotent ideals

Here is the question I'm working on: Show $R[x]$ is semiprimitive if and only if $R$ has no non-zero nilpotent ideals. Here's my attempt at $semiprimitive \Rightarrow no\space non-zero\space ...

1
2 3 4 5
12