Questions tagged [idempotents]

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

Let $S$ be a set endowed with a composition law $\cdot \colon S\times S\to S$. We say that $x$ is idempotent if $x\cdot x=x$.

429 questions
55
votes
13 answers

How to show that every Boolean ring is commutative?

A ring $R$ is a Boolean ring provided that $a^2=a$ for every $a \in R$. How can we show that every Boolean ring is commutative?
Paul
39
votes
5 answers

Proving: "The trace of an idempotent matrix equals the rank of the matrix"

How could we prove that the "The trace of an idempotent matrix equals the rank of the matrix"? This is another property that is used in my module without any proof, could anybody tell me how to prove this one?
Quixotic
  • 21,425
  • 30
  • 121
  • 207
33
votes
7 answers

Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a non-empty finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? Theorem 2.2.1. [R. Ellis] Let $S$ be a…
31
votes
2 answers

Are idempotent matrices always diagonalizable?

How to prove that any idempotent matrix is diagonalizable?
Lao-tzu
  • 2,658
  • 1
  • 20
  • 33
30
votes
5 answers

Idempotents in $\mathbb Z_n$

An element $a$ of the ring $(P,+,\cdot)$ is called idempotent if $a^2=a$. An idempotent $a$ is called nontrivial if $a \neq 0$ and $a \neq 1$. My question concerns idempotents in rings $\mathbb Z_n$, with addition and multiplication modulo $n$,…
A.B
  • 1,466
  • 16
  • 28
19
votes
5 answers

How many idempotent elements does the ring ${\bf Z}_n$ contain?

Let $R$ be a ring. An element $x$ in $R$ is said to be idempotent if $x^2=x$. For a specific $n\in{\bf Z}_+$ which is not very large, say, $n=20$, one can calculate one by one to find that there are four idempotent elements: $x=0,1,5,16$. So here is…
user9464
18
votes
4 answers

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 have $e(1 - e) = 0$. This is a special case of the…
user231212
14
votes
2 answers

Families of Idempotent $3\times 3$ Matrices

I did the following analysis for $2\times2$ real idempotent (i.e. $A^2=A$) matrices: $$ \begin{bmatrix}a&b\\c&d\end{bmatrix}^2=\begin{bmatrix}a^2+bc&(a+d)b\\(a+d)c&bc+d^2\end{bmatrix}=\begin{bmatrix}a&b\\c&d\end{bmatrix} $$ So in particular we have…
String
  • 17,222
  • 3
  • 38
  • 78
13
votes
1 answer

Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$

Let $\cal A$ be the (noncommutative) unitary $\mathbb Z$-algebra defined by three generators $a,b,c$ and four relations $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$. Is it true that $ab\neq 0$ in $A$ ? This question is natural in the context of an older…
12
votes
2 answers

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 only assume sufficiently many (but not…
Prism
  • 9,652
  • 4
  • 37
  • 104
11
votes
1 answer

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. $A_k = (A_k)^2$, $1 \leq k \leq m$, $A_kA_l=0$, for $k\neq l$, $1\leq…
FASCH
  • 1,632
  • 1
  • 16
  • 28
10
votes
1 answer

The number of esquares of idempotents in the rank 2 $\mathcal{D}$-class of $M_n(\mathbb{Z}_2)$.

Here's a variation of a question I was given during a research internship. Some Definitions: Definition 1: Let $S$ be a semigroup. For any $a, b\in S$, define Green's $\mathcal{L}$-relation by $a\mathcal{L}b$ if and only if $S^1a=S^1b$ and define…
Shaun
  • 38,253
  • 17
  • 58
  • 156
10
votes
1 answer

Commutativity of a ring from idempotents.

In a ring $R$ with unity, every element can be written as product of finitely many idempotents. Can one show that the ring is commutative?
9
votes
2 answers

Find all roots of $\,x^2\!\equiv x\pmod{\!900}$, i.e all idempotents in $\Bbb Z_{900}$

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 the answers by making a C++ program to test all…
9
votes
2 answers

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 $a \neq 0$ and $b\neq 0$. Is $R$ isomorphic to the…
Makoto Kato
  • 39,693
  • 9
  • 96
  • 219
1
2 3
28 29