# 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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### 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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### $n$-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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### 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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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 $$R:=\Bbb C[X_1,X_2,X_3,\dots]/\langle ... 1answer 22 views ### Nilpotents in a domain? What are the nilpotent elements in a domain? Doesn't it depend on what ring it is? Example, in \mathbb{R} it is 0 but in \mathbb{Z}_8 it is 2 because (2)^3 = 8 = 0. 2answers 84 views ### Prime ring with zero divisors Has Nilpotent Undergraduate Algebra by Matej Bresar, Exercise 2.104: Prove that a prime ring with zero divisors contains a non-zero nilpotent. Attempts: By the definitions, \forall a, b \in R-\{0\}, \exists x \... 1answer 133 views ### Asymptotic probability P(A^2=0) for a random matrix over \mathbb F_2 Suppose we choose a random n\times n matrix A over \mathbb F_2 by setting each entry of the matrix equal to 0 with probability 0.5 and 1 with probability 0.5 (this is equivalent to ... 1answer 36 views ### The relation of \alpha having index of nilpotence k>0 and being (with \sigma_{1}) 1-1 and onto. let V be a vector space over a field Fand let \alpha \in End(V) be nilpotent, having index of nilpotence k>0.Show that \sigma_{1} + \alpha \in Aut(V). where \sigma_{c} is defined as \... 0answers 23 views ### index of nilpotency of second derivative I have a linear operator over the complex field that is defined as the second derivative of x and y$$ \frac{\partial}{\partial x^2} + \frac{\partial}{\partial y^2}  V is the vector space of all ...
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 $k \neq 0$ ?