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 $x\cdot x=x$.

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

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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

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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

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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

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How to prove that any idempotent matrix is diagonalizable?

Lao-tzu

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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

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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

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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

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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

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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

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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

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[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

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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

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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?

Luckyluck63

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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

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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

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