# Questions tagged [idempotents]

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

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### If $M=\bigoplus_{i\in I}M_i$ and $f:M\to M$ is idempotent then $f$ induces an isomorphism between $M_i$ and $f(M_i)$.

Let $R$ be a ring, $M=\bigoplus_{i\in I} M_i$ an $R$-module with $\text{End}(M_i)$ local for all $i\in I$. Then I want to show that if $f:M\to M$ is idempotent (i.e. $f^2=f$) and nonzero, then there ...
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### Show whether the scalar projection is idempotent

A while ago I asked Show whether the vector projection is idempotent, which made it clear to me that the idempotency of the vector projection could be checked with a simple substitution. Now I am ...
1 vote
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### Finding idempotent elements of a $C^{*}$-algebra

Let $T \in B(\mathcal{H})$ (bounded operators on a complex Hilbert space $\mathcal{H}$) and suppose $T$ is normal. Suppose furthermore $\sigma(T)=\{-1\}\cup[2,3]$. Then I want to find a complete list ...
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### Find the determinant of a sum of idempotent matrices

Let $A, B \in \mathcal{M}_n (\mathbb{R})$ such that $A^2 = A$ and $B^2 = B$. If $\det(2A + B)=0$, prove that $$\det(A + 2B) = 0$$ My attempt: I know that both matrices are diagonalizable and their ...
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### Idempotent relations in a ring

Let $(A,+,.)$ be a ring such that, if $x \in A$ with $6x = 0$, then $x=0$. Let $a,b,c \in A$ such that $a-b$ , $b-c$ , $c-a$ are idempotent. Prove that $a=b=c$. Unfortunately, I haven't made any big ...
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1 vote
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### Checking primitivity of central idempotents: does it suffice to only consider central summands?

Let $R$ be a (not necessarily commutative) ring with $1$, and let $e$ be an idempotent, i.e. $e^2=e$. We call $e$ primitive if it is not the sum of two nonzero orthogonal idempotents. Assume that $e$ ...
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1 vote
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### Prove: $\det(I+A) = 2^{\text{rank}(A)}$ if $A$ is a square idempotent matrix. Find $(I+A)^{-1}$ such that the expression doesn't have inverses.

To prove: $\det(I+A)$ = $2^{\operatorname{rank}(A)}$ if $A \in$ $\mathbb{R}^{n\times n}$ and $A^{2}=A$. Find an expression for $(I+A)^{-1}$ such that it does not involve inverses. Is there any way I ...
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### Jordan decomposition of Idempotent matrix.

Matrix A $\in$ $\mathbb{R}^{n\times n}$ is idempotent if $A^{2} = A$. Describe the Jordan form of A. How do I do this? I am able to decompose a matrix to its Jordan form given that the matrix contains ...
1 vote
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1 vote
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### Conjugate idempotents

Let $R$ be a ring, and let $1 =e_1 +\cdots+ e_r = e'_1 + ...+ e'_r$ be two decompositions of $1$ into sums of orthogonal idempotents. If $e_1\cong e_i'$ for all $i$, show that there exists $u \in U(R)$...
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### About artinian rings and homomorphisms

Let $R$ be a left artinian ring, $\mathfrak{r}:=J(R)$. and e,f idempotent in $R$. Proof that the morphism of abelian groups $\varphi:eRf\rightarrow Hom_R(Re,Rf)$, $\varphi(erf)(r'e):=r'erf$ is an ...