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

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1answer
23 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 ...
3
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1answer
41 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 ...
5
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1answer
51 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 ...
2
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0answers
56 views

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
18 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 ...
1
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0answers
36 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$ ...
1
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1answer
20 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
24 views

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 ...
0
votes
2answers
27 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 ...
4
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2answers
33 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
25 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 ...
6
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1answer
69 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: ...
0
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1answer
168 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 ...
3
votes
0answers
37 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 ...
0
votes
1answer
69 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 ...
0
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1answer
30 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 $x \Delta y \Delta x=x$, ∀$x,y \in \mathbb{E}$.
2
votes
1answer
46 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. ...
3
votes
1answer
101 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 $X$. ...
2
votes
2answers
82 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 ...
0
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1answer
107 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?
1
vote
1answer
86 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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0answers
50 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 ...
2
votes
1answer
42 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 ...
1
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1answer
38 views

Idempotent and zero divisors

If $x^2=x$ and $x$ is non-unit then $x$ is a zero divisor in a ring $R$. I am trying to prove the contrapositive statement, that is Suppose $x$ is not a zero divisor and trying to show that $x$ ...
2
votes
1answer
112 views

A counterexample to $Ae\simeq Ae^{\prime}\to A(1-e)\simeq A(1-e^{\prime})$

Let $e,e^{\prime}$ be two idempotents in a $k$- algebra $A$ ($k$ is a field) . Then my guess $Ae\simeq Ae^{\prime}$ (as a left $A$- module) does not imply $A(1-e)\simeq A(1-e^{\prime})$ in general, ...
0
votes
1answer
29 views

Demonstrate that the centering matrix is idempotent

I want to demonstrate that the centering matrix $H$ is idempotent (i.e. $HH=H$). The centering matrix is defined as $H=I-\frac{1}{n}1\, 1^T$. I've tried developing this: $$ HH=\\ H(I-\frac{1}{n}1\, ...
1
vote
1answer
49 views

Proofing that an matrix is idempotent

My task was to show that certain matrices are idempotent, that is, ${AA} = {A}$. I struggled a with the proof for one case and when I allok at the solution, I have problems understanding onse step. ...
1
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0answers
115 views

Idempotents in quotient ring

Let $R$ be the ring $\mathbb{Z}[X,Y,Z]/(Y − X + 1, Y − Z + 2, 3X^2 - YZ + 3X +2Y + 4)$. What are the solutions $e \in R$ of the equation $e^2 = e$ with $e \not\in \{0,1\}$? So $e$ has to be a ...
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3answers
76 views

Find an invertible non-diagonal $3 \times 3$ matrix $D$ such that $D^3 = D$.

Find an invertible non-diagonal $3 \times 3$ matrix $D$ such that $D^3 = D$. I have forgotten how to solve this kind of question, can somebody give me some hint or idea how to start?
13
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2answers
310 views

Families of Idempotent $3\times 3$ Matrices

I did the following analysis for $2\times2$ real idempotent (i.e. $A^2=A$) matrices: $$ ...
4
votes
2answers
138 views

Idempotent elements of a ring.

I need the idempotent elements of $Z_{900}$ $2^2\cdot 3^2\cdot 5^2=900$ Of course there's $$0 \pmod 4 \\ 0 \pmod 9 \\ 0 \pmod {25} \\ $$ and $$ 1 \pmod 4 \\ 1 \pmod 9 \\ 1 \pmod {25} \\ $$ I found ...
1
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2answers
66 views

Given $P$ idempotent, show that $I-P$ is idempotent.

As title, given $V$ as a finite dimensional vector space over $F$. Suppose the linear transformation $P: V \rightarrow V$ is idempotent. I need to show that $I-P$ is idempotent. I know that a matrix ...
0
votes
1answer
168 views

Idempotent operators.

Apologies first. I am a physicist and my notations and rigour is probably lousy. If $P$ is an idempotent operator, $P^2 = P$, $P\neq \mathbb1$ and we have $\forall |\psi\rangle$ the relation, $P.L ...
1
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2answers
112 views

Idempotent operators over the exterior algebra

I am wondering if there exists a (reasonably) well-known set of operators $A_i$ over the exterior algebra such that $\{A_i,A_j\} = \frac{1}{2}(A_i +A_j)$, where $\{X,Y\}=(XY+YX)/2$.
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0answers
59 views

Relation between hyperbolic numbers and hyperbolic functions

Is there any relation between the hyperbolic (split-complex) numbers and hyperbolic trig functions? Or are they just named similarly by accident?
1
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1answer
134 views

Ring with non-trivial idempotent splitting as product of two rings

If a commutative ring $T$ has a non-trivial idempotent $e$, then it is easy to show that $f:T \rightarrow R\times S$ defined by $f(t)=(et,(1-e)t)$ is a ring isomorphism. My question is how to prove ...
2
votes
2answers
74 views

Idempotent Elements of a Commutative Ring

I have to prove this statement and I'm a bit unsure how to go about it: Show that the set of all idempotent elements of a commutative ring is closed under multiplication. Furthermore, find all the ...
1
vote
2answers
116 views

Square Idempotent matrix: efficient algorithms for finding eigenvectors

Given a square idempotent $N \times N$ matrix $A$ with large $N$, and a priori knowledge of the rank $K$, what is the most efficient way to compute the $K$ eigenvectors corresponding to the $K$ ...
5
votes
1answer
115 views

Spectrum of idempotent element

Let $A$ be some unitary algebra over $\mathbb{C}$. If $a^2=a$ and $0\ne a\ne 1$ then $\{0,1\}\subset \sigma_A(a)$ ($\sigma_A(a)$ is the spectrum if $a$). I believe that also $\sigma_A(a)\subset ...
0
votes
3answers
186 views

Showing that the only idempotents in $R$ are zero and one

I have the following question that I have to solve however I cannot achieve. Let $R$ be a ring with $1$ and suppose $R$ has no zero divisors. Show that the only idempotents in $R$ are $0$ and $1$. ...
2
votes
3answers
216 views

How to prove idempotent element is nilpotent

I have a problem that I need to solve but I have trouble in solving the following question. Question is; Let $a \in R$ be a nonzero idempotent. Show that $a$ is nilpotent. ($R$ is a ring) I ...
4
votes
2answers
183 views

Can an idempotent matrix have complex eigenvalues?

Let $P\in\mathbb{R}^{n\times n}$ be a nontrivial idempotent matrix: $P^2=P$, $P\neq 0$, $P\neq I,$ where $I$ is the $n\times n$ identity matrix. What are the eigenvalues of $P$? Solution: Let $x$ be ...
1
vote
2answers
93 views

co-idempotents: algebraic dual of an idempotent element?

So many times you can write out the axioms for an algebraic structure (say an algebra over a ring) as commutative diagrams and then reverse all the arrows and get a new structure: say a coalgebra. ...
2
votes
1answer
89 views

Are the two prime ideals containing same idempotents always the same?

If two prime ideals contain the same non trivial idempotents, what can we say about those ideals? Are they equal?
2
votes
1answer
47 views

Sum of primitive idempotents

Let $A$ be a finite dimensional commutative algebra over a field $K$ which can be written as a direct product of simple algebras $A\cong \prod_{i=1}^r A_i.$ My question: Is it possible to write ...
7
votes
1answer
192 views

Idempotents in a ring without unity (rng) and no zero divisors.

Question: Given a ring without unity and with no zero-divisors, is it possible that there are idempotents other than zero? Def: $a$ is idempotent if $a^2 = a$. Originally the problem was to ...
0
votes
0answers
58 views

Show $\mathbb F_{p}[x]/(x^{p}-1)$ is indecomposable as a representation of $\mathbb Z/ p\mathbb Z$

Let $R=\mathbb F_{p}[x]/(x^{p}-1)$. $R$ has both ring and vector space structure. I am trying to show that, given a representation $\rho : \mathbb Z/ p\mathbb Z\rightarrow GL(R)$, any invariant ...
3
votes
2answers
69 views

uniqueness of split idempotent

In a category, if an idempotent $f:a \longrightarrow a$ splits, then any two splittings are isomorphic. Let $i: b \longrightarrow a$ and $p: a \longrightarrow b$ be such that $i\circ p=f$ and $p\circ ...
3
votes
1answer
104 views

category theory, idempotent, splitting

In a small category $\cal{A}$, if a functor $F$ in $[\cal{A}^\text{op},\mathbf{Set}]$ is a retract of representable functor (i.e., a functor isomorphic to hom-functor),and $\cal{A}$ considered as a ...