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 $x\cdot x=x$.
Questions tagged [idempotents]
520 questions
60
votes
14 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
48
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
- 22,817
39
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…
user59671
36
votes
4 answers
Are idempotent matrices always diagonalizable?
How to prove that any idempotent matrix is diagonalizable?
Lao-tzu
- 3,066
34
votes
5 answers
Nontrivial idempotents in $\mathbb Z_n$ (roots of $x^2 = x$ and $\,x\neq 0,1)$
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
- 1,566
21
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
20
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
17
votes
3 answers
Does every commutative ring have $2^n$ idempotents?
I've spent a lot of time looking for examples, and I can't find any commutative rings which have a finite number of idempotents other than a power of $2$. Intuitively, adjoining an extra idempotent $a$ always seems to double the number, as for every…
Zoe Allen
- 7,939
15
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
- 18,838
14
votes
1 answer
Is there a difference between an idempotent function and a projection?
From what I can gather from the Wikipedia articles on idempotent function and projection, both terms refer to a function $f:S \to S$ such that $f \circ f \equiv f$, i.e. $f(x) = x$ for all $x$ in the image of $f$.
Is there any difference between…
tparker
- 6,756
13
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
- 11,782
13
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
- 1,762
- 1
- 21
- 31
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…
Ewan Delanoy
- 63,436
12
votes
3 answers
How to prove that $AB$ is a projection if $(AB)(BA)=AB$?
I was trying to solve the following problem:
Assume $A,B\in M_n\left( \mathbb{C} \right)$,satisfy $$AB^2A=AB.$$
I need to proof $$\left( AB \right) ^2=AB.$$
I tried to use some equivalent substitution of matrices, but I did not succeed. I also tried…
fusheng
- 1,238
11
votes
3 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…
Jack R
- 131