# Questions tagged [nilpotence]

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

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### Determine all $2 \times 2$ real matrices $A$ such that $(1) \ \ A^2=I$, $(2) \ \ A^2=0$

Determine all $2 \times 2$ real matrices $A$ such that $(1) \ \ A^2=I$, $(2) \ \ A^2=0$ I came across this problem recently where I have to determine all the $2\times2$ matrices satisfying the ...
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### Build a group $(G,+)$ from the nilpotent elements of the ring $(M_2(\mathbb{Z_2}),+,.)$, isomorphic with $(\mathbb{Z_2}),+)$. [on hold]

Build a group $(G,+)$ from the nilpotent elements of the ring $(M_2(\mathbb{Z_2}),+,.)$, isomorphic with $(\mathbb{Z_2}),+)$ and give two examples of nilpotent elements from$(M_2(\mathbb{Z_2}),+,.)$ ...
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### Nilradical is the intersection of finitely many minimal prime ideals.

Let $R \neq \{0 \}$ be a commutative ring with identity. Suppose that $R$ has only finitely many minimal prime ideals $p_1,\dots , p_s.$ Then $$\sqrt {0} = \bigcap\limits_{i=1}^{s} p_i.$$ I ...
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### Exericse about linear map $T\in L(V)$, where $\dim V=n\geq2$, with $\operatorname{null}T^{n-1}\neq\operatorname{null}T^n$

I have this problem that I am attempting, and am struggling with (b). -- Assume $\dim V = n \geq 2$ and that $T \in L(V)$ such that $\operatorname{null}T^{n-1}\neq\operatorname{null}T{^n}$ -- (a) ...
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### Nilpotent, Idempotent and Involutory Matrix

With exception of the zero matrix, can a matrix be nilpotent $(A^k=0)$ and idempotent $(A^2=A)$ at the same time? and With exception of the identity matrix, can a matrix be idempotent and involutory ...
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### Usage of Zorn's Lemma to prove that the intersection of all prime ideals contains only nilpotent elements.

I have read a couple proofs that that the intersection of all prime ideals contains only nilpotent elements that use a claim like this: Suppose that $a$ is an element of $A$ that is not nilpotent. ...
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### Proving that an element is nilpotent [closed]

Let $m,n \in \mathbb N, n \geq 2$. $A$ is a ring with $|A|=n$ and $a \in A$ s.t. $1-a^k$ is invertible $\forall k \in \{m+1,m+2,...,m+n-1\}$. Prove that $a$ is nilpotent. Can somebody help me, ...
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### k-th power nilpotence ($k\ge 3$)

Inspired by my solution to 2x2 Matrix with no zero entries where $A^k=0$ - Nilpotence? I came up with this problem. Let $A$ be a $nxn$ matrix which is not the zero matrix $0$ (in which all elements ...
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### 2x2 Matrix with no zero entries where $A^k=0$ - Nilpotence?

Find an example of a $2x2$ matrix $A$ that has no zero entries but is such that $A^K=0$ for some positive integer k. Here is my thinking: When $k=1, A=0$, but this contradicts that the matrix has no ...
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### Let B be a nilpotent linear operator with index p. Is operator $N^2 + \alpha*N$ nilpotent?

Let B be a nilpotent linear operator with index p. Is operator $N^2 + \alpha*N$ nilpotent? ( $\alpha \in \mathbb{R}$) I think that when $\alpha = 0$ then the index is $p/2$ for even p, but when p ...
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### Eigenvalues of a matrix whose square is zero

Let $A$ be a nonzero $3 \times 3$ matrix such that $A^2=0$. Then what is the number of non-zero eigenvalues of the matrix? I am unable to figure out the eigenvalues of the above matrix. P.S.: how ...
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### Show that two nilpotent matrices are closed under addition and multiplication

From Serge Lang's linear algebra textbook: A square matrix $A$ is said to be nilpotent if $A^r=O$ for some integer $r \geq 1$. Let $A$, $B$ be nilpotent matrices, of the same size, and assume $AB$...
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### If $G$ is a nilpotent group of class $2$, show $G' ≤ Z(G)$

If $G$ is a nilpotent group of class $2$, show $G' ≤ Z(G)$, and here we are taking the definition that class of nilpotency is the smallest length of the central series of $G$; where $G'$ is the ...
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### Show that if $R$ is a ring with a single ideal prime minimal $\neq(0)$ then $R$ has non-trivial nilpotent elements

Show that if $R$ is a ring with a single ideal prime minimal $\neq(0)$ then $R$ has non-trivial nilpotent elements, that is, elements to $a\neq 0$ for those who $a^n=0$, for a certain power $n> 0$. ...
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### Nilpotent elements of group algebra $\Bbb CG$

Goal: explicitly find a nilpotent element of the group algebra $\Bbb C G$ for some finite group $G$. This exists if and only if $G$ is non abelian by Maschke's theorem and Wedderburn-Artin. By ...
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### “Determine the zero divisors, nilpotents and the non-invertibles in $\mathbb{Q}[x] /(x^{4}-2x^{3}+3x^{2}-4x+2)$”

Good evening, I would like to ask the following exercise : "Determine the zero divisors, nilpotents and the non-invertibles in $\mathbb{Q}[x] /(x^{4}-2x^{3}+3x^{2}-4x+2)$" I saw that the ...
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### Nilpotent binary matrices over finite fields

I am studying on nilpotent matrices over finite fields. By definition a square matrix $A$ is $p$-nilpotent if a power of $A$ modulo $p$ is the zero matrix. For example. Let $J_2$ be the $2\times 2$ ...
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### Nilpotent Jacobson radical

I have to prove the following: If $R$ is a finite-dimensional algebra over a field $F$, then $J(R)$ is nilpotent. I thought about this, but there are some gaps: Because $R$ is in particular a ...
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### when the finite nilpotent group is cyclic?

In the notion of nilpotent group, a finite group $G$ is nilpotent iff every sylow subgroups is nomral in $G$ iff $G$ is isomorphic to direct products of all sylow subgroups of $G$ I wonder in what ...
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### If $\not\exists r+I$ nilpotent then $R/I$ is an integral domain

Let $R$ be a commutative ring and $I$ an ideal of $R$. Definition: a non zero element $a\in R$ is nilpotent if $\exists n\in\Bbb N$ s.t. $a^n=0$ It is true that if $R/I$ is an integral domain then ...
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### Let R be a prime Jacobson ring (with 1) [closed]

How to show that R has no nonzero nil ideals? Any hint would be great.
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### Prove that this linear transformation is nilpotent

The Question: Let $V$ be a finite dimensional vector space over $\Bbb C$, with linear map $T:V \rightarrow V$. Suppose that the minimal polynomial of $T$ is m_T(x) = p(x)q(x) \\ p(x)=(x-\lambda)^\...
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### Can the operator $\operatorname{id}_V-(AB-BA)$ be nilpotent in the infinite-dimensional case.

Let $V$ be a infinite-dimensional vector space over the field of characteristic $0$ and $A,B$ be linear operators of $V$. Let $\operatorname{id}_V$ be an identical operator. Using trace function it ...
$V$ is a vector space, $g: V \to V$ an isomorphism and $h: V \to V$ a nilpotent linear map. Also, $g$ and $h$ commute. This implies that $g+h$ is an isomorphism. Well, $h$ is nilpotent so $h^n=0$. ...
Let $R$ be a commutative ring with identity and put $U=U(n,R)$, the group of $n\times n$ (upper) unitriangular matrices over R. Define $U_i$ to be the be the subgroup of $U$ having (at least) $i-1$ ...