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

learn more… | top users | synonyms

1
vote
2answers
32 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 $K$ ...
4
votes
1answer
41 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
148 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
98 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
106 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
54 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
82 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
25 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
72 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
36 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
51 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
51 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
49 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
92 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
125 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
35 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
99 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
58 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
70 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
38 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
130 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
96 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
119 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
36 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
271 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
219 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
241 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
193 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
161 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
108 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
103 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.
4
votes
2answers
99 views

Idempotents in a Quotient Ring

Let $R=\mathbb{Z}_p[x]/(x^p-x)$. Show that $R$ has exactly $2^p$ elements satisfying $r^2=r$. I know that for $f,g\in\mathbb{Z}_p[x]$, we have $f-g\in(x^p-x)$ if and only if $f(a)=g(a)$ for all ...
1
vote
2answers
60 views

Is there a name for a relationship like idempotence between two functions?

If $f(f(x)) = f(x) \quad \forall \space x$ then $f$ is idempotent. If $g(f(x)) = f(x) \quad \forall \space x$ then is there a term to describe the relationship between $g$ and $f$?
7
votes
1answer
227 views

Sum of idempotent matrices is Identity

[Ciarlet, Problem $1.1-10$] Let $A_k$, $1 \leq k\leq m$, be matrices of order $n$ satisfaying $$\sum_{k=1}^mA_k\ =\ I.$$ Show that the following conditions are equivalent. $A_k = ...
1
vote
1answer
137 views

Formal Notation of Functions

Say I have a function $f$ with the following signature: $f: S \times S \to S$ and I wanted to formally represent that this function was idempotent and reflexive. How would i do right this? For ...
3
votes
0answers
94 views

Representation for idempotent semiring

I have a semi-ring whose multiplication is non-commutative, and addition is idempotent. That is, $ab \neq ba$ and $a + a = a$. The semi-ring is freely generated from a finite set $\Sigma$, the ...
0
votes
0answers
119 views

A special idempotent function

Usually an idempotent function is $f: S \rightarrow S$ such that $f \circ f = f$. But I need something slightly different: $f: S \times S \rightarrow S$, such that: For $x, y, z \in S$: $f(x, y) ...
0
votes
1answer
1k views

Determine if the matrix is idempotent?

I am dealing with an example to show that the matrix($M = I − X(X'X)^{−1}X'$) is idempotent. X is a matrix with T rows and k columns and I the unit matrix of dimension T. And then to determine the ...
4
votes
1answer
97 views

Do Ramsey idempotent ultrafilters exist?

I was studying idempotent ultrafilters when I saw that no principal ultrafilter could ever be idempotent, because $\left\langle n \right\rangle \oplus \left\langle n \right\rangle = \left\langle 2n ...
2
votes
0answers
44 views

Left continuous magmas with no fixed points

Let $X$ be a compact Hausdorff topological space, and $*: X^2\rightarrow X$ an associative map (so that $(X, *)$ is a semigroup) which is left continuous (for all $s\in X$, the map $t\mapsto ts$ is ...
0
votes
0answers
57 views

fully idempotent and co-idempotent [closed]

I want the proof of : For each prime number p, the sub module N = Zp ⊕0 of the Z-module M = Zp⊕Zp is not idempotent but N = Hom z (M;N)N. Z4 is a multiplication Z-module which is not fully ...
5
votes
2answers
180 views

Can a sum of idempotents vanish?

Let $A$ be a finite dimensional $\mathbb C$-algebra. Let $e_1,\ldots,e_r\in A$ be nonzero idempotents (with $r>0$), i.e. $e_i^2=e_i$. My question is: Can it happen that $e_1+\cdots+e_r=0$? I can't ...