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

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2
votes
2answers
56 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
votes
1answer
18 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
31 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?
1
vote
0answers
42 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
28 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
vote
1answer
30 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 $x$ ...
1
vote
1answer
88 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
24 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
39 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
vote
0answers
80 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 ...
-1
votes
3answers
68 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
votes
2answers
295 views

Families of Idempotent $3\times 3$ Matrices

I did the following analysis for $2\times2$ real idempotent (i.e. $A^2=A$) matrices: $$ ...
-3
votes
2answers
85 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
vote
2answers
50 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
87 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
vote
2answers
101 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 $\{X,Y\}=(XY+YX)/2$.
0
votes
0answers
36 views

Relation btwn 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
vote
1answer
73 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
49 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
77 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
67 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
166 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
147 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
164 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
79 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
88 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
36 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 ...
3
votes
1answer
141 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
49 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
65 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
76 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 ...
0
votes
1answer
59 views

Is there a standard name for a set equipped only with an idempotent binary operation?

Is there a name for an idempotent magma, or do they not arise often enough to warrant a special name? (By idempotent binary operation, I mean an operation $+$ such that $x + x = x$ for any $x$.)
3
votes
4answers
127 views

Rings with zero divisors in which $x^2=x$ implies $x\in\{0,1\}$

Any idempotent element (other than 0 or 1) must be a zero-divisor, so in particular there are no nontrivial idempotents in domains. Are there examples of rings with zero divisors without ...
3
votes
2answers
140 views

$r$ is not nilpotent, $r-r^2$ is nilpotent, then the ring has a non-zero idempotent

Assume that $R$ is a ring and $r-r^2$ is nilpotent for an element $r\in R$. If $r$ is not nilpotent, then $R$ has a nonzero idempotent.
1
vote
2answers
84 views

Idempotency and Jordan cells

Which Jordan cells $J(\lambda,k)$ are idempotent? And how can I use that to determine the Jordan canonical form of any square idempotent matrix?
-2
votes
2answers
127 views

Show that an Integral Domain contains exactly two idempotents. [closed]

An element $a$ of a ring is an idempotent if $a^2=a$. Show that an Integral Domain contains exactly two idempotent elements. I don't know where to start. Any help/hints would be greatly ...
3
votes
3answers
95 views

conditions for idempotence in $2 \times 2$ matrix

Let $A=\begin{bmatrix} a&b\\c&d\\\end{bmatrix}$. I'm looking for the conditions such that $A^2=A$. So I start calculating... $$A^2=\begin{bmatrix} ...
1
vote
2answers
85 views

Prove that an idempotent $e$ of $R$ is primitive iff $\dim_{\,F}\left(\operatorname{Im} e\right)=1$

Let $V$ be a vector space over the field $F$, $R$ is the ring of linear operators on $V$. Prove that an idempotent $e$ of $R$ is primitive iff $\dim_{\,F}\left(\operatorname{Im} e\right)=1$ Thanks a ...
2
votes
1answer
44 views

Two idempotent matrix

Let $A,G$ be two $n\times n$ matrix satisfying: $$A^2=A, GAG=G, im(G)\subset im(A).$$ Prove that $G^2=G$. I do not know how to prove it.
3
votes
3answers
172 views

A problem on nilpotent matrix and commutator

Let $A$ be a $n\times n$ nilpotent matrix, that is, for some $m\geq 1$, $A^m=0$. Suppose $\operatorname{rank}(A)=n-1$, and define a map from $M_n(\mathbb{C})$ (the complex matrices) to itself by ...
2
votes
0answers
164 views

Cyclic linear codes and idempotents

Got this assignment from coding class and would be very thankful for checking if my solutions are correct. a) Find all idempotents modulo $1 + x^{17}$ of degree at most $15$ So first i find $r$ from ...
3
votes
1answer
124 views

Idempotents in $M_2(\mathbb{C})$

Given two idempotents $e,f\in M_2(\mathbb{C})\setminus\{I_2\}$, the sets $$\{eg^{-1}:g\in GL_2(\mathbb{C}), eg^{-1} \text{ is an idempotent}\}$$ and $$\{gf:g\in GL_2(\mathbb{C}), gf \text{ is an ...
2
votes
0answers
38 views

$\mathbb{Z}/\mathbb{nZ}$ admits idempotents [duplicate]

I need to show that if $n$ is not a prime power,then $\mathbb{Z}/\mathbb{nZ}$ admits idempotents $\neq 0,1$ I noticed this thing for $\mathbb{Z_6}$ and $\mathbb{Z_{12}}$ and few more but how do ...
7
votes
3answers
357 views

The structure of a Noetherian ring in which every element is an idempotent.

Let $A$ be a ring which may not have a unity. Suppose every element $a$ of $A$ is an idempotent. i.e. $a^2 = a$. It is easily proved that $A$ is commutative. Suppose every ideal of $A$ is finitely ...
2
votes
1answer
323 views

Prove that every idempotent element is not nilpotent element.

Let $R$ be a ring. Prove that every idempotent element is not nilpotent element. I've got a problem with proving this question. I was be grateful, if somebody would be help me.
3
votes
2answers
333 views

Find all idempotent elements in the group algebra $\mathbb CC_3$

Sorry but I'm quite new to group algebras and even Latex so if this is all wrong I apologize. By $\mathbb CC_3$ I mean the group algebra of the cyclic group of order 3 in the complex numbers A group ...
8
votes
2answers
199 views

Characterization of the field $\mathbb{Z}/2\mathbb{Z}$

Let $R \neq 0$ be a ring which may not be commutative and may not have an identity. Suppose $R$ satisfies the following conditions. 1) $a^2 = a$ for every element $a$ of $R$. 2) $ab \neq 0$ whenever ...
8
votes
2answers
209 views

Sufficiently many idempotents and commutativity

It is a well-known result that if a ring $R$ satisfies $a^2=a$ for each $a\in R$, then $R$ must be commutative. See here for proof. I am wondering whether the same result holds for finite rings if we ...
5
votes
1answer
132 views

Idempotents in rings without unity

Suppose there are non-trivial idempotents in the ring without unity. Is it right that all of them are zero divisors? If we're given unitary ring with unity $e$ and $a$ is non-trivial idempotent ...
1
vote
1answer
140 views

Selfadjoint operator $\Rightarrow$ Idempotent Operator?

If $P\in\mathcal{L}(H,H)$, with $H$ a Hilbert space, such that $P = P^*$, Is possible to show that $P^2 = P$? If that is possible, then $P$ is a projection operator, right? Thanks in advance.