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

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Idempotents which are not Mouray von neumann equivalent to its adjoint

What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Mourray Von neumann equivalent to $e^{*}$?
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1answer
15 views

Why is the Young symmetrizer non-zero?

Suppose $\lambda$ is a partition of the natural number $n$ and $T$ is a standard Young Tableaux of shape $\lambda$. Let $$P_{\lambda}:=\lbrace g\in S_n:g\text{ preserves the rows of }T\rbrace$$ and ...
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2answers
42 views

Prove that the sum of ideals of a ring A equals A and its intersection is zero.

I've been looking at a couple of ring theory exercises and there's this one I don't know how to do it. It goes like this. $A$ is a commutative unital ring, and $e$ an element of $A$, $e \neq ...
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0answers
22 views
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Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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3answers
33 views

Show that if $\rho$ is idempotent then $\rho$ acts as the identity on $\rho(V)$

A linear map $V \xrightarrow{\rho} V$ is idempotent if $\rho\rho = \rho$. Show that if $\rho$ is idempotent then $\rho$ acts as the identity on $\rho(V)$. (Such linear maps are called projections: ...
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1answer
21 views

eigenvalues of composite matrix

I have a $n$ x $n$ symmetric positive definite matrix $C$, and a symmetric idempotent matrix $M$, given by $I-\frac{1}{n}J_n$ where $J_n$ is a $n$ x $n$ matrix of all ones. Additionally, all the ...
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1answer
100 views

K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
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0answers
41 views

nil Jacobson radicals

Let $f$ be an idempotent element in a ring $S$ with Jacobson radical $J$ so that both $fJf$ and $(1-f)J(1-f)$ are nil. I guess that $J$ is nil too, but I am not sure. I know that the former is ...
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0answers
40 views

Homomorphisms of modules over a corner ring

Let $R$ be a Noetherian ring and suppose that we can write $1 = e_1 + e_2 + \dots + e_n$ where the $e_i$ are pairwise orthogonal idempotents. Let $S = e_1 S e_1$, and consider the right $S$-modules ...
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2answers
38 views

Idempotent $17\times 17$ Matrix Jordan Normal Form

I have a $17\times 17$ idempotent matrix $M$ with rank $10$ $M$ is idempotent so $M^2=M \rightarrow M(M-1)=0$. So the minimum polynomial is $x(x-1)$. The eigenvalues of $M$ are $x=0$ and $x=1$. The ...
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2answers
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If $A,B,A+I,A+B$ are idempotent matrices how to show that $AB=BA$?

If $A,B,A+I,A+B$ are idempotent matrices how to show that $AB=BA$ ? MY ATTEMPT: $A\cdot A=A$ $B\cdot B=B$ $(A+I)\cdot (A+I)=A+I$ or, $A\cdot A+A\cdot I+I\cdot A+I\cdot I=A+I$ which implies ...
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1answer
41 views

Idempotent ideals having only idempotents

I search for sufficient conditions for a ring $R$ so that any idempotent ideal constitutes only of idempotent elements of $R$. Of course, in the commutative case, any finitely generated idempotent ...
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2answers
30 views

Nonminimal idempotents

Let $S$ be a semigroup with operation $+$. Note that we do not assume $S$ is abelian. We say that $I\subset S$ is a left ideal if and only if for all $s\in S$, $s+I\subseteq I$, and we say that an ...
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0answers
34 views

S an inverse semigroup with semilattice of idempotents E and $\sigma$ the minimum group congruence on S.

Let S be an inverse semigroup with semilattice of idempotents E, and let $\sigma$ be the minimum group congruence on S. Show that the following statements are equivalent: (a) $x\sigma y$; (b) ...
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2answers
22 views

Find scalars that result in an idempotent matrix

For what scalars, $b$ and $c$ will the following matrix be idempotent: $bI_m + c1_m1_m^T$ I know that idempotency implies that each eigenvalue is 0 or 1, but I am still having trouble finding the b ...
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1answer
28 views

Prove $a,b\in R$ are idempotent iff $ab=0$.

Let $R$ be a ring. An element $a\in R$ is called idempotent if $a^2=a$. Suppose that R is a ring with unity $1\neq 0$, and that there are $a,b\in R$ such that $a+b=1$. Prove that $a,b$ are idempotent ...
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4answers
134 views

Proving that if $AB=A$ and $BA=B$, then both matrices are idempotent

Let $A, B$ be two matrices such that $AB=A$ and $BA=B$, how do I show that $A\cdot A=A$ and $B\cdot B=B$? Steps I took: Let $A= \left[\begin{array}{rr} a & b \\ c & d \\ ...
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2answers
70 views

Does inverse image of idempotent element contain an idempotent element? [closed]

Suppose $f\colon \oplus A_i\to\oplus B_j$ is a ring homomorphism, where $A_i,B_j$ are local rings. Both sides have finitely many summands. Suppose an idempotent $x$ (i.e. $x=x^2$) is in the image of ...
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2answers
67 views

Let $A ∈ M_{n×n}(\mathbb{R})$ differ from $I$ and $O$. If $A$ is idempotent, show that its Jordan canonical form is a diagonal matrix.

Let $A ∈ M_{n×n}(\mathbb{R})$ differ from $I$ and $O$. If $A$ is idempotent, show that its Jordan canonical form is a diagonal matrix. I'm not sure how to do this. Any solutions/hints are greatly ...
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2answers
45 views

Finding (orthogonal) idempotents in a quotient of a polynomial ring

I wish to identify all of the non-trivial idempotents in the ring $$ \Bbbk[x]/\langle x(x+1)^2(x+2)^2(x+3)\rangle.$$ I have reason to suspect that there are at least four which are, in addition, ...
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2answers
51 views

Convex hull of idempotent matrices

What is the convex hull of the set of $n\times n$ (potentially asymmetric) idempotent matrices? Apparently the powers that be want more information: Consider the set $S:=\{A\in\mathbb{R}^{n\times ...
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1answer
63 views

Idempotent projection operators in a Hilbert space

Let $H$ be a Hilbert-space and $S$ be a sub(Hilbert)space such that: $$H = S \oplus S^\perp$$ Then the projection operators are defined as: $$P_S: H\to S; x = u + v \mapsto u \quad\quad P_{S\perp}: ...
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1answer
65 views

The $e$ in the operator $exe'$

I am learning the group algebra. A way to study the irreducible representations of a finite group is to use the idempotent operators in the group algebra. Suppose $e$ and $e'$ are the primitive ...
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4answers
378 views

Intuition for idempotents, orthogonal idempotents?

Given a ring $A$, an element $e \in A$ is called an idempotent if one has $e^2 = e$. If $e$ is an idempotent, then so is $1 - e$, since$$(1 - e)^2 = 1 - 2e + e^2 = 1 - 2e + e = 1 - e.$$Also, we ...
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1answer
103 views

Idempotents in commutative ring

I have proven for $e$ an idempotent in a commutative ring $R$ that there exists an isomorphism $\phi:R\rightarrow R/eR\times R/(1-e)R$ given by $r\mapsto(r,r-re)$. Now I would like to prove that ...
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2answers
38 views

Idempotent in a ring of continuous functions?

Let $S$ be a set of continuous functions defined on an interval $[0,1]$. The addition and multiplication between two elements is defined as $(f+g)(x)=f(x)+g(x)$ and $(f\cdot g)(x)=f(x)g(x)$. So $S$ ...
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1answer
32 views

Show that $R\cong Re\times R(1-e)$

An element $e\in R$ is called idempotent if $e^2=e$. Assume that $e$ is idempotent and $er=re$ $\forall r\in R$. Prove that $Re$ and $R(1-e)$ are two sided ideals of $R$ and $R\cong Re\times ...
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1answer
66 views

If $R$ is a simple ring, is every corner $eRe$ simple?

Assume that $R$ is a ring, not necessarily commutative nor unital. Let $e$ be an idempotent in $R$. Is there a correspondence between ideals of $R$ and ideals of the corner ring $eRe = \{ere : r\in ...
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1answer
52 views

Question concerning a minimal faithful left ideal in an artin algebra

Let $B$ be an artin algebra an suppose there is a faithful projective-injective left $B$-module. Moreover, there is a minimal faithful left ideal $Be$ for some idempotent $e\in B$. 1) What does ...
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Idempotent semiring

Let $R$ be a semiring. For $a\in R$,we define $t_a(x)=x+a$ for all $x\in R$. Prove that $R$ is idempotent(with +) and $1$ has an infinite order if and only if for all $a,x,y\in R$, ...
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0answers
32 views

Ellis semigroup

I have only seen the construction of the Ellis semigroup applied to a group endowed with the discrete topology. Does the same idea works for any topological group? If not, where does it go wrong? (I ...
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0answers
37 views

Determine all ring homomorphisms from $ \Bbb Z$ $\oplus$ $\Bbb Z $ to $\Bbb Z$. [duplicate]

I got $(a,b) \to a$, $(a,b) \to b$ and $(a,b) \to 0$ these mappings to be homomorphisms just by hit and trial. So when I looked for it's solution, these were the ONLY homomorphisms from $ \Bbb Z$ ...
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1answer
22 views

Find Matrix $S$ such that $S'CS=\left(\begin{array}{cc} 1_{N-1} & 0 \\ 0 & 0 \end{array} \right)$ where $C:=1_N-\iota \iota'$

The Centering Matrix $C:=1_N-\iota \iota'$ has eigenvalue $1$ of multiplicity $n − 1$ and eigenvalue $0$ of multiplicity $1$. Therefore a matrix $S$ with columns consisting of eigenvectors of $C$ can ...
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0answers
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Find basis of Image and Kernel of Linear Transformation

I asked earlier about proving the Idempotency of a Linear Transformation. The question was: Suppose that $a_1,...,a_n,b_1,...,b_n ∈ F$ are such that $􏰀a_ib_i = 1_F$. Let $J : F_n → F_n$ be the ...
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2answers
34 views

Proving Idempotency for general matrix

Suppose that $a_1,...,a_n,b_1,...,b_n ∈ F $ are such that 􏰀$\sum a_ib_i = 1_F$. Let $J : F^n → F^n $ be the linear transformation whose standard matrix has $ij^{th}$ entry $a_ib_j$. Prove that ...
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2answers
87 views

Given a ring with unity and a central idempotent element e, prove some isomorphic relations

Given a ring $R$ with 1 $\neq$ 0, and an element $e$ that is idempotent and central in $R$, I want to prove that $R/Re \cong R(1-e)$, $R/R(1-e)\cong Re$, and subsequently, $R\cong Re\times R(1-e)$. My ...
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0answers
39 views

Matrix representations and idempotent/nilpotent elements

A conceptual question: Let's say there's a linear matrix representation of a particular algebra. I'm wondering just how much this matrix representation can tell us about the 'outstanding' elements ...
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1answer
77 views

Given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute.

Let $R$ be a ring with identity. An element $a \in R$ it is idempotent if $a^2=a$. Show that given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute. Remark: ...
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1answer
187 views

Proof of Lallement’s Lemma

i am a bit weak with my congruence manipulation when it comes to semigroup theory. Could you possibly check my proof and give me constructive criticism on it? Lemma: Let $S$ be a regular semigroup ...
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0answers
46 views

Idempotents and symmetries of Zn

[Soft question out of curiosity] While reading about idempotents of a ring, I used $\mathbb Z_n$ as a convenient example. In visualizing the ring's structure, I was intrigued by strange symmetries ...
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1answer
120 views

If $V$ is a vector space to itself, is the idempotent linear transformation, prove there is a basis such that is a diagonal matrix with all $0$ or $1$

Im trying to solve an algebra homework problem, basically i have: $$\phi:V \longrightarrow V$$ such that $\phi = \phi^2$, where $V$ is finite dimensional. I already proved that $V = \phi(V) \oplus ...
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0answers
18 views

References on projectors

What are good books or articles about linear projectors in Hilbert spaces? I am mostly interested in the finite dimensional case (but anything is welcome). All about idempotents, orthogonal and ...
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1answer
45 views

Associative Binary Operation(composition) is anticommutative iff idempotent…

if Binary Operation, $\Delta$, defined on $\mathbb{E}$ is associative, then $\Delta$ is anticommutative iff $\Delta$ is idempotent and $x \Delta y \Delta x=x$, ∀$x,y \in \mathbb{E}$.
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1answer
50 views

For which $n \in \mathbb{N}$ is it the case that every element of $\mathbb{Z}/n\mathbb{Z}$ is strongly associate to an idempotent?

Definition. Call two elements of a commutative ring associates iff each divides the other. Call them strong associates if there exists a unit that can be multiplied by the first to yield the second. ...
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1answer
111 views

If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents

Let $R$ be a commutative ring with identity, $P$ be a prime ideal in $R$ and define $$X := \lbrace t \in P \mid t^2=t \rbrace. $$ Also let $J$ denote the smallest ideal of $R$ that contains $X$. ...
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2answers
102 views

Find idempotents in $\Bbb Q[x]/\langle x^2 - 1\rangle$

I know that in $\Bbb Z[x]/\langle x^2 - 1\rangle$ the trivial idempotents are $0,1$ and the other idempotents are those elements in $\Bbb Z[x]$ that have the remainder $0$ or $1$ when divided by $x^2 ...
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1answer
276 views

general idempotent matrix possible values of the determinant

If A is a general idempotent matrix, calculate the possible values of det (A) I caculated the det = o what other values can it equal?
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1answer
192 views

Prove that in a finite monoid each element is invertible

Let $(M,\circ)$ be a finite monoid. Suppose the identity element $e\in M$ is the only idempotent element. Then prove that each element in $M$ has inverse. How can I prove this?
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59 views

Idempotent generators of the four binary QR codes of length 7

I have a coding theory assignment and I thought it would be a good idea to double check before I hand it in. I'm asked to find the idempotent generators of the four binary QR codes C1, C2, C3, C4, of ...
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1answer
50 views

Suppose that $R$ is a commutative ring with unity, $a \in R$, and $\varphi(r) = ar$ defines a ring homomorphism. Prove that $a$ is idempotent.

Suppose that $R$ is a commutative ring with unity, $a \in R$, and $\varphi(r) = ar$ defines a ring homomorphism $\varphi$. Prove that $a$ is idempotent, i.e. that $a = a^{2}$. This is exercise 15 ...