# Questions tagged [nilpotence]

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

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### Looking for a counterexample when dropping one of the constraints (Linear algebra)

I want to find a counterexample for the following "Theorem": Let $V \neq 0$ be a finite dimensional $K$ - vector space and $L \subset \mathfrak{gl}(V)$ a linear subspace. If all $x \in L$ ...
1 vote
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### If R is a ring with idempotent then eR is clean ring if and only if eRe is clean ring

I am reading this paper on clean rings: Clean general rings, by W.K. Nicholson, Y. Zhou, and I am puzzled by the proof of Corollary 11: Corollary 11. If $R$ is a ring and $e^2=e \in R$, then $eR$ is ...
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### reduction of matrix (nilpotence)

I'm struggling doing this exercise: Let A,B be two matrices in M_n(R) where N is nilpotent and NA = 0_n. We must show that the characteristic polynomial of A+N is exactly the one of A. Well, since I ...
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### For nilpotent matrix $N$ over complex field, and for natural number $r$ show that there exists $A$ in $M_n(\mathbb{C})$ s.t. $A^r = N + I_n$ [closed]

I have been given as an exercise the following statement to prove, but I have no idea where to start and how to go about proving it. If $N$ is a nilpotent matrix in $M_n(\mathbb{C})$ and $r$ is a ...
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1 vote
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### Does $(x^2)(x^3) \neq 0$ imply $(x)(x^3) \neq 0$?

Consider a commutative unital algebra $A$ of finite dimension $n>3$ over the reals. The product is defined such that elements are generated with real number coefficients $(a_0, \dotsc, a_n)$ for ...
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### Is the span of all nilpotent ideals also a nilpotent ideal?

Given a non-zero Lie algebra $\mathcal{L}$ over $\mathbb{C}$, we define $\mathcal{L}^2 = \big[\mathcal{L}, \mathcal{L} \big] = \big\{ [x, y]: x, y\in \mathcal{L} \big\}$, and for any $k\in\mathbb{N}$ ...
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1 vote
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### index of nilpotency of a linear transformation

Let $\Delta : M \in \mathcal{M}_n(\mathbb{C}) \mapsto AM+MB$ with $A,B \in \mathcal{M}_n(\mathbb{C})$. Q1) Show that if $A$ and $B$ are nilpotent matrix then $\Delta$ is nilpotent Q2) What is the ...
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### 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{...
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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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1 vote
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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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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 ...