# Questions tagged [idempotents]

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

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### Incorrect proof for $e$ idempotent $\Rightarrow$ $eA$ projective

I have seen several proofs for $e$ idempotent $\Rightarrow$ $eA$ projective where $A$ is an algebra. I tried something different and produced a proof without using the fact that $e$ is idempotent (so ...
1 vote
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### Two comaximal radical ideals, with product the nilradical, are radicals of principal ideals generated by complementary idempotents

Let $\mathfrak a,\mathfrak b\subset R$ be two comaximal radical ideals of the ring $R$, with $\mathfrak{ab}=0$. Then $\mathfrak a=Ra$ and $\mathfrak b=Rb$ for complementary idempotents $a,b\in R$. The ...
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### Is an idempotent logical expression considered to be in conjunctive normal form?

Is $$A \lor B \lor A$$ technically in conjunctive normal form? Or must we apply the idempotent law for it to be in CNF? $$A \lor B \lor A \equiv A \lor B$$?
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### Prove that for every idempotent element a belonging to $A$ (ie such that $a^2 = a$) and for every $b$ belonging to $A$ then $ab = ba$.

Let $A$ be a ring without nonzero nilpotent elements. Prove that for every idempotent element a belonging to $A$ (ie such that $a^2 = a$) and for every $b$ belonging to $A$ then $ab = ba$. I know that ...
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### A nice characterization of commuting idempotent endomaps?

Let $X$ be a set, and let $f\colon X\to X$ be a map. It is idempotent when for each $x\in X$, $f(f(x))=f(x)$. It is equivalent to the requirement that the restriction of $f$ on its image $f(X)$ is the ...
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### Relations, Transitive functions & Idempotent functions

I've read all the stack threads I can find on transitive functions and want to pull together the ideas/questions it leaves me with. This all started because as per this thread I wrongly thought ...
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### How can I prove that this matrix is idempotent?

I have the following matrix $$A=\begin{equation} \begin{pmatrix} 0 & a & -b\\ -a & 0 & c\\ b & -c & 0 \end{pmatrix} \end{equation}$$ I have to prove that $M=A^2+I$ is ...
1 vote
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### Idempotent matrix $X^2=X$ with complex entries

Suppose I have an Idempotent matrix (with complex entries) $X \in \mathbb{C}^{n \times n}$ such that $X^2 = X$ and I want to determine the set of solutions of this equation. So, I started by ...
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Let $R$ be an associative ring with unity $1$ generated by all its idempotents. Denoted by $R^n$ the additive subgroup of $R$ generated by all elements $a_1a_2\cdots a_n$ for $a_i\in R$. Then, is $R=R^... 0 votes 1 answer 71 views ### If$\mathcal{C}$is a cyclic code, with generating idempotent$ε(x)$, then$ε(x)-θ(x)$is a generating idempotent for$\mathcal{C}_0$We consider a cyclic code$\mathcal{C}$over a finite field$\mathbb{F}$, with generating idempotent$ε(x)$and$\mathcal{C}_0$a subcode, which contains all the (code)words of zero sum. If$θ(x) = 1/... First of all, we consider a finite field $\mathbb{F}$, with $|\mathbb{F}|=q$, and a natural number $n$ that is relatively prime to $q$. We also consider the quotient ring $\mathcal{R}_n = \mathbb{F}[... 3 votes 3 answers 379 views ### Assistance with idempotent matrices I am taking linear algebra for the first time and am struggling with the concept of idempotent matrices. I know that$A = A^2$is the concept behind it, but I can't seem to understand HOW one would ... 2 votes 1 answer 92 views ### Idempotents in tensor product with a f.g. local Artin algebra Let$k$be a field. Let$A$and$B$be$k$-algebras (commutative , with unit). Assume that$A$is finitely generated, local and Artinian. Assume that the set of idempotents in$B$is finite. (I stress ... 13 votes 3 answers 399 views ### How to prove that$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 ... 2 votes 1 answer 92 views ### How to proof there is no idempotent element other than 0 and 1 in a Division Algebra? If$A$is a division$K$-algebra. Then I need to proof there is no idempotent element other than$0$and$1_A$in$A$. I tried this way : If$0,1_A\neq a\in A$such that$a^2=a.~$Now$A$is division ... 1 vote 3 answers 103 views ### Let$e$be an idempotent of the monoid$M$,$x$,$y$be two elements of$eMe$. Then,$(eMe)\,x\,(eMe) = (eMe)\,y\,(eMe)$if and only if$MxM = MyM$[duplicate] I am studying semigroups. I saw a Lemma in the text that states: Let$e$be an idempotent of the monoid$M$,$x$,$y$be two elements of$eMe$. Then,$(eMe)\,x\,(eMe) = (eMe)\,y\,(eMe)$if and only ... 7 votes 1 answer 114 views ### Does this property of algebra morphisms (related to idempotents) have a name? Let$F$be a field. I am in the category of finite-dimensional$F$-algebras. Let$f:A \rightarrow B$a homomorphism of two of those. The property of$f$which came up as useful in something I consider ... 2 votes 1 answer 76 views ### Is it possible to characterize the set of$k$-periodic matrices$\in \mathbb{R} ^{n \times n}$? A$k$-periodic matrix$M$is an$\mathbb{R}^{n \times n}$matrix such that$M^{k+1} = M$Is it possible to characterize this set of$k$-periodic matrices? My goal is to exploit this characterization ... 2 votes 1 answer 67 views ### If$A$has no non-trivial idempotents, then neither does$A/N$Let$A$be a commutative, associative, unital, finitely generated algebra over an algebraically closed field$k$. Denote by$N$the nilradical of$A$, which is the set of all nilpotent elements of$A$... 2 votes 2 answers 120 views ### Non-commutative abelian rings. Let$R$be a ring with unity. An element$e\in R$is called idempotent if$e^2=e$. Clearly,$0,1$are idempotents. An element$e\in R$is called central if$er=re$for all$r\in R$. Recall that a ring ... 5 votes 0 answers 119 views ### If$M=\bigoplus_{i\in I}M_i$and$f:M\to M$is idempotent then$f$induces an isomorphism between$M_i$and$f(M_i)$. Let$R$be a ring,$M=\bigoplus_{i\in I} M_i$an$R$-module with$\text{End}(M_i)$local for all$i\in I$. Then I want to show that if$f:M\to M$is idempotent (i.e.$f^2=f$) and nonzero, then there ... 0 votes 0 answers 21 views ### Show whether the scalar projection is idempotent A while ago I asked Show whether the vector projection is idempotent, which made it clear to me that the idempotency of the vector projection could be checked with a simple substitution. Now I am ... 1 vote 1 answer 119 views ### Finding idempotent elements of a$C^{*}$-algebra Let$T \in B(\mathcal{H})$(bounded operators on a complex Hilbert space$\mathcal{H}$) and suppose$T$is normal. Suppose furthermore$\sigma(T)=\{-1\}\cup[2,3]$. Then I want to find a complete list ... 0 votes 2 answers 115 views ### Find the determinant of a sum of idempotent matrices Let$A, B \in \mathcal{M}_n (\mathbb{R})$such that$A^2 = A$and$B^2 = B$. If$\det(2A + B)=0$, prove that $$\det(A + 2B) = 0$$ My attempt: I know that both matrices are diagonalizable and their ... 3 votes 1 answer 75 views ### Idempotent relations in a ring Let$(A,+,.)$be a ring such that, if$x \in A$with$6x = 0$, then$x=0$. Let$a,b,c \in A$such that$a-b$,$b-c$,$c-a$are idempotent. Prove that$a=b=c$. Unfortunately, I haven't made any big ... 2 votes 1 answer 52 views ### Invertible indempotent matrix not equal to identity On Wikipedia, I found that the only non-singular idempotent matrix is$\mathrm{\mathbf{I}}$. However, the following matrix$$\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0 \\ 0 & 1 & 1\... 1 vote 1 answer 38 views ### Idempotent Semigroup$S$with Equivalence Relation$(a R b) \iff (aba=a), (bab=b)$:$S/R$is commutative - why? This is exercise 11.19 (h) in Seth Warner's "Modern Algebra". We are given that$S$is an idempotent semigroup, that is:$\forall a \in S: aa = a$. Let$R$be the equivalence relation ... 0 votes 1 answer 745 views ### Let$𝐴 $be a square matrix of order$𝑛$such that$A^2=A$Prove that every$𝑣∈\mathbb R^n$can be decomposed as$ 𝑣=𝑣_1+𝑣_2$[closed] Let$A$be a square matrix of order$n$such that$A^2 = A$. Prove that every$v \in \Bbb R^n$can be decomposed as$v = v_1 +v_2$, where$v_1$is in the null space of$A$and$v_2$is in the column ... 2 votes 1 answer 68 views ### Conditions for Group given Semigroup with Idempotent Element Another exercise in Seth Warner's "Modern Algebra" (1965). This one is Exercise 7.15. Let$(S, \circ)$be a semigroup. Let$(S, \circ)$have an idempotent element$e$, that is, such that$e \...
E.g there is a formula for a function that gives a number of idempotent functions for a set with finite size. The solution is $\sum_{k=1}^n{n\choose k}k^{n-k}$ but you need to sum intermediate results....