Questions tagged [idempotents]

For questions about elements which satisfy $x\cdot x=x$ where $\cdot$ is a composition law.

492 questions
Filter by
Sorted by
Tagged with
1 vote
11 views

The residue class of a complete set of primitive orthogonal idempotents

I was studying the book Elements of the Representation Theory of Associative Algebras: Volume 1 and this question occurred to me: In page 29, it says that Because $\{e_1,…,e_n\}$ is a complete set of ...
1 vote
22 views

• 2,287
16 views

Primitive non-central idempotents of a group algebra

Let $W_i;\ 1 \le i \le m$ be irreducible $\mathbb{C}$-representations of a finite group $G$ with $\mathbb{C}$-characters $\chi_i$. Let $(V,\rho$) be a $\mathbb{C}$-representation of $G$ with isotypic ...
• 2,287
49 views

• 2,287
1 vote
47 views

When are Idempotents elements of a semisimple algebra primitive

Let $A=KG$ be a $K$-algebra such that $|G| \in K^{\times}$. Here $A$ is a semisimple algebra. Consider the decomposition of $A$ into simple components:$$A=A_1 \times A_2 \times \cdots \times A_k.$$ ...
• 2,287
1 vote
26 views

Primitive idempotent and bilateral ideals

I'm trying to show for my algebra class that in a semisimple ring with unity $R$ (not necessarily commutative), every primitive idempotent element must belong to a minimal two-sided ideal. Here, by ...
• 321
116 views

Dimension of the center of a block

Let $G$ be a finite group and let $F$ be an algebraically closed field of characteristic $p$. I've been studying some modular character theory from Navarro's "Characters and blocks of finite ...
• 2,663
17 views

primitive idempotent

I’ve tried to prove: $R$ local, Artin ring $\phi:End(P) \rightarrow End(Q)$ ,$e\in End(P)$ is primivite idempotent and $P$ and $Q$ free module so $P= (\oplus R_i)$ and $Q=(\oplus R_j)$then how ...
37 views

Von Neumann regular rings and its ideals

Let R be a strongly regular Von Neumann ring, this is, that for every $r \in R$ there exists $x \in R$ such that $r^2x=r$. From here, how to prove that $R$ is strongly Von Neumann regular if and only ...
• 121
57 views

• 343
154 views

about $\operatorname{tr}(A) = \operatorname{rank}(A)$ for idempotent matrix $A$

I'm trying to prove the above statement, and I had a look at this site. This ends the proof with the following statement the rank is the number of non-zero eigenvalues But, this is what makes me ...
• 769
1 vote
109 views

Prove $\lVert A \rVert _{p} \geq 1$ for any idempotent matrix $A \neq 0$

As the title states, I'm facing a problem to prove: for any idempotent matrix $A \in \mathbb{C}^{n \times n}$ and $A \neq 0$, $\lVert A \rVert_{p} \geq 1$. Here the p-norm $\lVert A \rVert_{p}$ ...
• 13
65 views

75 views

Are idempotents always central in a von Neumann regular ring? [closed]

A ring $R$ is called von Neumann regular if for every $a \in R$; $a=axa$ for some $x \in R$. A ring $R$ called Abelian if every idempotent in $R$ is central. Are von Neumann regular rings abelian?
• 47
70 views

• 48
1 vote
128 views

If $R$ is commutative and $I$ is a finitely generated ideal with $I^2=I$, then there exists an idempotent $e\in I$ with $I=Re$

Question Let $R$ be a commutative ring. Let $I$ be a finitely generated ideal. Assume that $I^2=I$. Show that $I$ is a direct summand of $R$. Answer I know that $I=Re$ for some idempotent $e\in I$ and ...
• 489
316 views

How to prove $\mathbf{1}^\top\mathbf{Q}^+\mathbf{Q}=\mathbf{1}^\top$, where $\mathbf{Q}$ is any element-wise squared correlation matrix?

Let $(X_1,…,X_n)$ be a random vector with $0<\prod_{j=1}^n\text{Var}(X_j)<∞$. Let $\mathbf{Q}=(\mathbf{q}_{1},…,\mathbf{q}_{n})=(ρ_{jk}^2)_{n×n}$, where $ρ_{jk}$ is the Pearson correlation ...
• 87
80 views

Does there exist an idempotent, pseudo-constant $p$-adic function with an uncountable image?

$f$ is defined to be a pseudo-constant function if $f'(x)=0$. The question simply comes from idly wondering, "What are p-adic idempotent functions like?" Differentiable idempotent functions ...
• 2,597
1 vote
49 views

Orthogonal idempotents in rings without unity?

Some context In this answer and comments, an argument is given for why an ordered ring with unity can't have nontrivial idempotents. I'm trying to extend the argument for an ordered ring $R$ without ...
• 1,249
113 views

Find all idempotent matrices such that $(A-B)^2 = 0$

Find all idempotent matrices such that $(A-B)^2 = 0$ We can see that the hypotheses imply that $A+B=AB+BA$, and if we multiply by $AB$ on the right, we get $AB+BAB=(AB)^2+BAB$, which also implies ...
• 585
1 vote
251 views

• 147
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 ...
Let $X$ be a set, and let $f\colon X\to X$ be a map. It is idempotent when for each $x\in X$, $f(f(x))=f(x)$. It is equivalent to the requirement that the restriction of $f$ on its image $f(X)$ is the ...