Questions tagged [nilpotence]

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

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$ab=0 \Rightarrow b$ is nilpotent in a Noetherian ring

Let $A$ be a Noetherian ring and $a \in A$. Show that if $a$ is not contained in a minimal prime, $ab=0 \Rightarrow b$ is nilpotent. I can't see a way to solve this. I tried to consider that in a ...
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Show that two complex nilpotent matrices are similar [duplicate]

We consider he following proposition. Two $n\times n$ complex nilpotent matrices are similar if and only if they have same nilpotence index. How do I show that the statement is correct for $n=3$ ? ...
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How can I construct a nilpotent matrix of order 100 and index 98?

I know to construct a nilpotent matrix of order $n$ with index of nilpotency $n$, but how to construct a nilpotent matrix of order $n$ but index of nilpotency $(n-2)$? Is there any general rule for ...
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$A^n=0$ if $a_{ij}=0$ for all $i\geq j$.

Prove: $A^n=0$ if $a_{ij}=0$ for all $i\geq j$ where $A\in M_n(K)$. I tested it for a few matrices and I guess I have to do it by induction. How can I do that?
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Matrices Question invlving transpose

Could anyone solve this question using properties? I did it by taking a $2\times1$ matrix.
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Is there a matrix $A \neq 0$ such that $A\in F^{2\times 2}$ and $A^2=0$?

Any hints? i don't know how to disprove the statement I looked at the multiplication with parameters and looked at the different cases but there were not enough information $F$ is a field
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Classifying certain types of matrices

How many similarity classes of nilpotent $4 \times 4$ matrices over $\mathbb{C}$ are there? I suspect the answer is connected to minimal polynomials, but I'm not sure. Any suggestions?
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Examples of locally nilpotent ring

Let rings be associative, but not necessarily unital or commutative. We say that a ring is locally nilpotent if every finitely generated subring is a nilpotent ring. I am interested in finding some ...
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Decomposition of vector space to produce nilpotent and invertible transformations [duplicate]

Let $V$ be a finite-dimensional vector space over a field $K$. Let $T$ be a linear operator on $V$. Prove that there exists a unique sum $V=V_{0}+V_{1}$ such $T(V_{0}) \subseteq V_{0}$, $T(V_{1}) \...
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Base change of nilradical is nilradical

Let $B \to A, B \to B'$ be injective, finite ring homomorphisms (finite means that $A$ and $B'$ are finite $B$-modules). Suppose that $A$ and $B$ are integral domains. Denote by $N$ the nilradical of ...
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Finding Basis of Nilpotent

Consider the differential operator $T:P3(R)→P3(R)$ given by $T(p(x))=2p′(x)−2p′′(x)+2p′′′(x)$ Find an ordered basis $F$ for $P3(R)$ such that $T$ acts like a shift operator with respect to ...
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If ${\rm Im} (T) = \ker (T)$, then $T$ is nilpotent.

I have to prove, that if ${\rm Im} (T) = \ker (T)$, then the transformation matrix is nilpotent. How can I do this? I know the Rank–nullity theorem: If $T: V \to W$, then $\dim{\rm Im}(T) + \dim \...
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If $A$ is an $n \times n$ matrix such that $A^5 = 0$, which of the following is true?

I have this question from my textbook: Given $A$ an $n \times n$ matrix such that $A^5 = 0$, which of the following is true? $(a)$ $A$ is invertible. $(b)$ $A = 0$. $(c)$ $A$ is diagonalizable. $(...
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Prove that $S+T$ is nilpotent

Let $V$ be a $K$-vector space and let $S$ and $T$ be two linear operators of $V$. If $S$ and $T$ are both nilpotent and $ST=TS$ then $S+T$ and $ST$ are nilpotent operators I got this: By hypothesis, $...
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Lie algebra with $\mathfrak{g}/\mathcal{C}_2\mathfrak{g}$ not the free 2-step nilpotent Lie algebra

I am looking for examples of finite-dimensional complex nilpotent Lie algebras $\mathfrak{g}$ where the quotient $\mathfrak{g}/\mathcal{C}_{2}\mathfrak{g}$ of the Lie algebra with its 2nd lower ...
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Characterization of Complex Nilpotent matrices?

I know every complex matrix can be made into a triangular matrix and characterization of Nilpotent matrix. Is this a better characterization of Nilpotent matrices? I tried to show M is a Nilpotent ...
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Let $U$ be an $F$-invariant subspace such that $F$ is nilpotent on $U$. Then $U \subset \text{Ker } F^q$.

I am reading "Introduction to Linear Algebra" (in Japanese) by Kazuo Matsuzaka. There is the following problem in this book: Let $V$ be a finite-dimensional vector space. Let $F$ be a linear ...
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Prove that if a real matrix $A$ is symmetric and $A^2=0$, then $A=0$

Prove that if a real matrix $A$ is symmetric and $A^2=0$, then $A=0$. Didn't really understand how a squared matrix could be equal to $0$.
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Similar concept for “nilpotent” matrix but gives the identity and not 0?

Is there a term for a square matrix $E$ such that $E^k=I$ for some positive integer $k$? To provide context: I was experimenting with permutation matrices and discovered that they satisfy the ...
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Equivalent condition for a Lie algebra to be Nilpotent

In these notes, the author, N. Perrin, states that a Lie algebra $\frak{g}$ is nilpotent if and only if we can find a decreasing sequence $(\frak{g}_i)_{i=0}^n$ of ideals in $\frak{g}$ such that $\dim\...
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Find idempotents in a set of mappings

Find idempotent elements of $S$ the set of maps $f:X \rightarrow R$ where $X$ is a given set and $R$ is a given ring with the operations defined by $(f+g)(r)=f(r)+g(r)$ and $(fg)(r)=f(r)g(r)$ where ...
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Let M be invertible and N be nilpotent. Show that M-N is invertible

I have the following task: Let $M, N \in M_n(\mathbb{R})$ be two matrices s.t. $MN=NM$. (1) Show that if $M$ is invertible and $N$ be nilpotent of order k, that $M-N$ is invertible. (2) Let $M$ ...
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How would you find the characteristic polynomial of a nilpotent matrix with complex coefficients?

I'm trying to find the characteristic polynomial of an nxn nilpotent matrix with complex coefficients. I understand how to do it with standard coefficients but I'm a bit confused as to how to proceed ...
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Must the product of a nil ideal with a minimal right ideal be 0?

Let $R$ be a ring (not necessarily with identity or commutative). Suppose that $K$ is a a nil ideal, and let $M$ be a minimal right ideal of $R$. Must it then follow that $MK = 0$? As $M$ is minimal ...
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Prove that a non-zero matrix satisfying $A^N = \textbf0$ for some $N$ cannot be diagonalized.

Let a non-zero matrix $A$ satisfy $A^5 = \textbf0$. Prove that A cannot be diagonalized. More generally, any non-zero nilpotent matrix, i.e. a non-zero matrix satisfying $A^N = \textbf0$ for some $N$ ...
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In a non-commutative ring (possibly without identity) with no nontrivial automorphisms, do nilpotent elements form an ideal?

In an old exam appeared this statement: True/False: "Let $R$ be a ring with the property that the unique ring automorphism is the identity. Then the set of all nilpotent elements form an ideal". I'...
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Is an ideal where all elements have $x^m=0$ nilpotent?

Let $m$ be a positive integer, let $R$ be a (non-commutative) ring and let $I$ be a two-sided ideal of $R$ such that $x^m=0$ for all $x \in I$. Is there an integer $M$ such that $I^M=0$? Thoughts: ...
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Inverse of $I + A$

I am trying to solve the following exercise in Artin, without breaking into cases for even and odd $k$. A square matrix $A$ is called nilpotent if $A^k = 0$ for some $k > 0$. Prove that if $A$ ...
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In every local ring exist nonzero nilpotent ideals?

I read that every local ring is clean so there exist clean rings with nonzero nilpotent ideals, i know that a local ring has a unique maximal ideal, but i don't know why do they say this implication. ...
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Proving that $N(R)\times N(S)= N(R\times S)$

Let $R,S$ be rings with $1_R,1_S$ respectively. Thanks to this answer, I realised that I probably make a mistake in proving that $$N(R) \times N(S)=N(R\times S),$$ where $N(R)$ is the nilradical (...
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Strange result on the nilradical $N(R)$ of a ring

I am studying about the nilradical $N(R)$ of a unital ring $R$. In my notes, the nilradical of a $R$ is defined as the sum of all nilpotent ideals of $R$. It says, that $N(R)$ is always a nil ideal, ...
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Index of a Nilpotent matrix

I was wondering why there can't be a nilpotent matrix of index greater than its no. of rows. Like why there does not exist a nilpotent matrix of index 3 in $M_{2×2}(F)$ When I look up on the ...
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What can be said about the rank of a nilpotent matrix?

Is there anything that can be said about the rank of a nilpotent $n \times n$ matrix? It certainly is less than $n$ but anything else?
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Prove that $A$ and $B$ are nilpotent - proof checking

Let $A$ and $B$ be $n \times n$ matrices with real entries and $c_1, c_2, \dots ,c_{n+1}$ distinct real numbers such that $A+c_1B, A+c_2B \dots, A+c_{n+1}B$ are nilpotent matrices. Prove that $A$ and $...
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$n$-th power of a matrix [duplicate]

I came a cross a problem I could not solve today: If matrices $A$ and $B$ satisfy $A^4 = 0$ and $B=I-A$, prove that $$B^{-1}=I+A+A^2+A^3$$ I doubt they expect me to start calculating 4 matrices ...
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Examples for commutative rings with many nilpotent elements yet the nilradical is not

What is an example (if exists) for a commutative ring that contains many nilpotent elements (preferably even with the same exponent, i.e., there are $a_1,...,a_r\in \mathbb{R}$ with $a_i^k=0$), such ...
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Proving by induction that strictly lower triangular matrix is nilpotent

Let matrix $A \in \mathbb{K}^{n \times n}$ have the property $a_{ij} = 0$ for $1 \leq i \leq j \leq n$. Show that $A^n = 0$. Proof by induction: Base Case: for $n=2: A= \left[ {\begin{array}{...
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Give an example of an operator $T \in L(\mathbb C^7,\mathbb C^7)$ such that $T^2+T+I$ is nilpotent.

Give an example of an operator $T \in L(\mathbb C^7,\mathbb C^7)$ such that $T^2+T+I$ is nipotent. Attempt: If we choose $\lambda$ such that $\lambda^2+\lambda+1=0$ and define $T$ such that $T v = \...
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T ∈ L(V) is nilpotent if and only if σ(T) = {0}.

I know that T is nilpotent if and only if only eigenvalues of T is 0. How can we prove the statement $T ∈ L(V)$ is nilpotent if and only if $\sigma (T) = \{0\}$ where $\sigma (T)$ denotes singular ...
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Non-zero nilpotent operator

I know that if an operator $T\in L(V)$ is nilpotent, then $T^{dim(V)}=0$. I am struggling to construct an example of a nilpotent operator $T\in L(V)$ which satisfies the condition $T^{dim(V)-1}\neq 0$...
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Prove that there exist a diagonalizable operator $D$ and a nilpotent operator $N$

Let $\mathbb V$ be an $n$ dimensional complex vector space. If $T \in L(V,V),$ prove that there exists a diagonalizable operator $D$ and a nilpotent operator $N$ such that $T = D + N$ and $DN = ND$ ...
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$L_1,L_2$ linear and $L_1\circ L_2$ nilpotent $\implies L_2\circ L_1$ nilpotent

Given $L_1,L_2\in\mathcal{L}(V)$ and $L_1L_2$ nilpotent. I have to show that $L_2L_1$ ist nilpotent either. Let $L_1,L_2\in\mathcal{L}(V)$ and $L_1L_2$ nilpotent. Let $v\in V$. Than there $\exists m\...
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1answer
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On the Nilradical $N(R)$ of a noncommutative ring.

I am doing some Noncommutative Algebra and I am studying the notion of the nilradical of a (not necessarily) commutative ring. More precisely: Let $R$ be a ring with $1_R$. The nilradical $N(R)\...
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On the nil but not nilpotent ideal counterexample.

In this post, there is a counterexample where a nil ideal (ie every element is nilpotent) is not nilpotent. More precisely, let's consider the polynomial ring $$R:=\Bbb C[X_1,X_2,X_3,\dots]/\langle ...
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Nilpotents in a domain?

What are the nilpotent elements in a domain? Doesn't it depend on what ring it is? Example, in $\mathbb{R}$ it is 0 but in $\mathbb{Z}_8$ it is $2$ because $(2)^3 = 8 = 0$.
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Prime ring with zero divisors Has Nilpotent

Undergraduate Algebra by Matej Bresar, Exercise 2.104: Prove that a prime ring with zero divisors contains a non-zero nilpotent. Attempts: By the definitions, $\forall a, b \in R-\{0\}, \exists x \...
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Asymptotic probability $P(A^2=0)$ for a random matrix over $\mathbb F_2$

Suppose we choose a random $n\times n$ matrix $A$ over $\mathbb F_2$ by setting each entry of the matrix equal to $0$ with probability $0.5$ and $1$ with probability $0.5$ (this is equivalent to ...
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The relation of $\alpha$ having index of nilpotence $k>0$ and being (with $\sigma_{1}$) 1-1 and onto.

let V be a vector space over a field $F$and let $\alpha \in End(V)$ be nilpotent, having index of nilpotence $k>0.$Show that $\sigma_{1} + \alpha \in Aut(V).$ where $\sigma_{c}$ is defined as $\...
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index of nilpotency of second derivative

I have a linear operator over the complex field that is defined as the second derivative of x and y $$ \frac{\partial}{\partial x^2} + \frac{\partial}{\partial y^2} $$ V is the vector space of all ...
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Rank and Nilpotence

Is it possible to find a square matrix $M$ such that $M^N=0$ , $M^{N-1}\neq 0$ With $rk(M^n)=rk(M^{n+k})$, for some $n, n+k \lt N$ and $k \neq 0$ ?

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