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 $x \in L$ ...
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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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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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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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Algorithm and program for modelling a Free Nilpotent Lie algeabra

I need to compute in a Free Nilpotent Lie Algebra $L$ given by a finite list of generators. For example, put the generators $\{A, B\}$. So, the linear generators for the space of $L$ is $$\{A, B, [A,B]...
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Are there any non-trivial nilpotent and idempotent in the ring $F[X,Y]/(X^2-Y^2)$, where $F$ is a field?

I can only come up with the following: If there is a nontrivial nilpotent element $r+(X^2-Y^2)$, then $r^n\in(X^2-Y^2)$ for some integer $n$; and $r\not\in (X^2-Y^2)$ Similarly for a nontrivial ...
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Matrix with all off-diagonal elements equal 1

Consider the matrix, of size $n \times n$, of the following form: \begin{bmatrix} a_1 & 1 & \cdots & \cdots1\\ 1 & a_2 & 1 &\cdots 1\\ \vdots & 1 & a_3 &\cdots 1\\ ...
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Questions regarding the construction of nilpotent matrices in light of certain characteristics

I have some questions regarding nilpotent matrices. I know that the trace of an $n\ x\ n$ nilpotent matrix must be zero and that the rank of that matrix must be less than $n$. Thus, the determinant ...
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In a ring $A$ such that $a^3=a$ $\forall a \in A$, determine the nilpotent elements of $A$.

Let $a \in A$ be a nilpotent element. There is an integer $k \geq 1$ such that $a^k = 0$. Let $m$ be the minimum of these integers. By the euclidian division of $m$ with the number $3$, there exist ...
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If $A$ is nilpotent and $AB = A$, does $B$ have to be identity matrix?

I have seen posts asking under what conditions does $B$ have to be the identity matrix if $AB = A$, but none of them could answer my question. I know that if $A$ is singular, then $A = AB$ doesn't ...
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Extending the matrix property to nilpotent operators in infinite dimensional spaces. [closed]

Let $A \in M_n(\mathbb{C})$ be nilpotent matrix such that $A^{4} = 0$, We can find matrix $B \in M_n(\mathbb{C})$ such that we have $$AB \neq 0 \ \ and \ \ BA = 0$$ The question is as follows: How can ...
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Why is every element of a ring not a nilpotent?

An element $a$ of a ring $R$ is nilpotent if there is an integer $n$ such that $a^n = 0$. Every element in a ring must have some $n$ that makes this true (the order of the element). So why is every ...
Kevin DeCara's user avatar
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Nilpotence of a linear endomorphism

Let $F$ be a field, $V$ a finite-dimensional $F$-vector space and $f \in \operatorname{End}_{F}(V)$. Show: If $\mathbb{F}=\mathbb {C}$ and there is no one-dimensional subspace $U$ of $V$ with $ f(U)=...
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How to prove this Lie algebra is nilpotent? [duplicate]

Let $L$ be a Lie algebra over the field $F$ with the property that $\forall x,y,z\in L ; [[x,y],z]=0$ . How to show that $L^3=0?$ Here $L^3=[L,L^2],L^2=[L,L^1],L^1=[L,L].$ There is no condition on the ...
Bony Spectrum's user avatar
2 votes
1 answer
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Given a nilpotent matrix $A$ of index 3, prove that its transpose $A^T$ is also nilpotent of index 3

Given the matrix $A$ which is nilpotent of index 3 (which implies $A^3=0$ and $A^2\neq0$), and using matrix and nil-potency properties I have to prove that its transpose is also nilpotent of index 3 ...
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Prove if $M^2 =0$, then $\operatorname{rank}(M+M^T)=2\operatorname{rank}(M)$.

Prove if $M^2 =0$, then $\operatorname{rank}(M+M^T)=2\operatorname{rank}(M)$. I have used the rank nullity theorem and the fact that rank of matrix and its transpose is same to prove it but I am not ...
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Nilpotent, upper triangular sum and product of matrices

In the MSE question it was claimed that if matrices $AB$ and $A+B$ are nilpotent then $A,B$ are nilpotent. However generally the claim is false - user1551 has found quickly counterexample. I wonder ...
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conditions for an algebra generated by idempotents to be solvable/nilpotent

Given a finite Lie algebra that is generated by a set of rank-1 idempotents $\{P_{j}\}_{j}$ over the complex field, what are the conditions for it to be solvable (or alternatively nilpotent). A ...
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There is $f$ nonnilpotent in $A[[x]]$ with nilpotent coefficients -- solution verification [duplicate]

This is an exercise in Atiyah-MacDonald (Exercise 5.2). I have shown that if $f \in A[[x]]$ ($A$ commutative with unity) is nilpotent, then $f$'s coefficients are all nilpotent. The next question in ...
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How to prove that $2A+B^2$ is nilpotent matrix

I have $A,B \in \mathbb C^{3\times3}$ nilpotent matrix's. We can suppose that $AB^2=B^2A$, is it true that $2A+B^2$ is nilpotent? what i did is $(2A+B^2)^3$= $8A^3+12A^2B^2+6AB^4+B^6$ so now i know A,...
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Nilpotent elements in a quotient of a polynomial ring.

I'm trying to prove the following statement. I'm not entirely sure if it's true though I suspect it is: Consider the ring $A = \mathbb Z[x]/(f_1,\ldots,f_n)$. Then the element $x \in A$ is nilpotent ...
Jordan Levin's user avatar
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Let $A$ and $B$ be nilpotent $n \times n$ matrices with $AB = BA$. Show that $A + B$ is also nilpotent. [duplicate]

Because $A$ and $B$ are nilpotent we know that $A^n$ and $B^n$ are $0$. I guess we can rewrite $(A+B)^n$ as $\sum_{k=0}^n {n \choose k}A^kB^{n-k}$ and then use $AB = BA$ to rewrite this in a way that ...
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Prove that the following statements are equivalent.

Let A be a nxn Matrix whose characteristic polynomial XA can be decomposed into a product of linear factors. A^n = 0 A is nilpotent 0 is the only Eigenvalue of A. I did the steps 1) => 2) but I'm ...
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Proving an algebra is nilpotent

I recently have started studyng about free algebras and I want to know what kind of methods or approaches are there to prove that free algebra $A$ is nilpotent with nilpotency index $n.$ Let me remind ...
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Is every solvable algebras are nilpotent?

An algebra $A$ is called nilpotent if $A^n=0$ for some positive integer $n$. Also, we call algebra $A$ is solvable if $A^{(n)}=0$, with solvable index $n$, where $A^{(n)} = A^{(n-1)}A^{(n-1)}$. Can we ...
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Is the only possible multiplicative structure on a finite dimensional vector space that of a product ring of finite field extensions?

Let $K$ be a field and $R$ a commutative ring that's a finite dimensional $K$-vector space. Do we necessarily have (as rings) $$ R \cong \Pi_{i=1}^n F_i $$ for some finite field extensions $F_i \...
George's user avatar
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Nilpotent operator over even dimension real vector space

Prove that if V is a real vector space of even dimension 2k, then every operator T ∈ L(V ) such that $T^2 + T + I$ is nilpotent satisfies $(T^2 + T + I)^k = 0$ Hint: For a,c, you may want to argue ...
UnknownPlayer's user avatar
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2 answers
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Given $X, Y \in \Bbb C^{n \times n}$ and $XY+2YX=3I$, show $[X,Y]$ is nilpotent

Given $X, Y \in \Bbb C^{n \times n}$ and $XY+2YX=3I$, show that $[X,Y]$ is nilpotent. I tried with Jacobson Theorem but it does not work. Trying to calculate traces of powers by induction to see if ...
Stefan Solomon's user avatar
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why is a Lie algebra not nilpotent even if an ideal is nilpotent and algebra quotient by ideal is nilpotent [duplicate]

Why can't we use the same arguments as for solvability to say that If a Lie algebra $L$ has a nilpotent ideal $I$, and $L/I$ is nilpotent, then $L$ is nilpotent. Can someone point out the fallacy in ...
matzo's user avatar
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Endomorphism mod nilpotent ideal is identity imply isomorphic?

Let $R$ be a communicative unity ring, $I\subseteq R$ is a nilpotent ideal, $M$ is a $R$-module, $f:M\rightarrow M$ is a $R$-homomorphism. If the induced map $\bar{f}:M/IM\rightarrow M/IM$ is identity ...
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Simple Zorn Lemma application doubt.

I am currently going over the proof of the following proposition: (I know there is more than one post about this same result, but none can answer my current doubt). Propostion. Given a commutative ...
xyz's user avatar
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${\rm Spec}(A/\mathfrak{a})={\rm Spec}(A)$ if and only if $\mathfrak{a}$ is generated by nilpotent elements

I have been introduced to the Zariski topology and I cant solve this problem: Let $A$ be a commutative ring with unity and $\mathfrak{a}$ an ideal of $A$, we define ${\rm Spec}(A) $ as the set of ...
Pablo Borrego's user avatar
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1 answer
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Justify the use of Zorn's Lemma in a proof

I was reading this proof of how the nilradical of a ring is the intersection of all prime ideals of the ring. https://artofproblemsolving.com/wiki/index.php?title=Nilradical In the proof, it says &...
Michael Wang-Wakamatsu's user avatar
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Prove that for every idempotent element a belonging to $A$ (ie such that $a^2 = a$) and for every $b$ belonging to $A$ then $ab = ba$.

Let $A$ be a ring without nonzero nilpotent elements. Prove that for every idempotent element a belonging to $A$ (ie such that $a^2 = a$) and for every $b$ belonging to $A$ then $ab = ba$. I know that ...
Misimiausis's user avatar
3 votes
1 answer
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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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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 \...
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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)=\...
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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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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 $...
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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'...
funkypeanut's user avatar
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1 answer
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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}(...
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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 ...
Manish Saini's user avatar
2 votes
2 answers
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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 ...
Anton Shcherbina's user avatar
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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, ...
Hussain-Alqatari's user avatar
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1 answer
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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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1 answer
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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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