# Questions tagged [nilpotence]

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

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### 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 ...
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### 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 ...
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### 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'...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $lim_{t \to 0} P_t^{-1}XP_t = Y$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $T$ ...
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### 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 ...
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### $T \in \text{Hom}V$ 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 ...
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### 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 ...
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### 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 $0$ ...
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### 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 ...
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### 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 ...
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### $N$ 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 ...
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### 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 ...
### $R[x]$ 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 ...