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Questions tagged [idempotents]

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

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Prove that there's only one matrix which is involutory and idempotent at the same time.

An involutory matrix $\mathbf{A}$ is such that $\mathbf{A}^{2}=\mathbf{I}$. An idempotent matrix $\mathbf{A}$ is such that $\mathbf{A}^{2}=\mathbf{A}$ So we'd have to satisfy both conditions ...
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All idempotents splits in set theory context

So I wanted to show that in the category $\mathbf{SET}$, all idempotent splits. Fix an idempotent $f$, I wish to find 2 arrows $g,h$ such that $f=hg$ and $gh = 1$. I'm having some trouble "...
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Determine if a matrix is an orthogonal projection matrix

We define an orthogonal projection as a linear transformation that maps a vector into its orthogonal projection in some (given ahead) subspace $W$. Let's call the matrix of that transformation (...
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1answer
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Given P idempotent, show that I−P is idempotent. [duplicate]

So my task is well summed up by this older post: Given $P$ idempotent, show that $I-P$ is idempotent. PandaMan idea is that by proving $(I−P)^2 = (I-P)$ we prove that $(I-P)$ which implies that ...
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How do you prove there is a bijection between an isomorphism and a set of orthogonal idempotents?

Let $A$ be a commutative ring with unity. How do you prove there is a bijection between: An isomorphism as a product of rings $\phi: A \longrightarrow A_1\times\cdots\times A_n$ A decomposition as ...
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Idempotents in $ \mathbb{Z}_n $ [closed]

Let $ n=cd $ where $ c $ and $ d $ are co-primes. Then there are integers $ x $ and $ y $ such that $ xc+dy=1 $. How can it be proved that $ xc$ is idempotent in $ \mathbb{Z}_n $? Is converse ...
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Show that if A = B + C, then BC = 0

A,B,C are n $\times$ n symmetric and idempotent matrices. The question is: If $$ \textbf{A = B + C } $$ then show $$ \textbf{BC = 0 } $$ I'm not sure where to start?
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matrix algebra and idempotent matrix

I'm having a little trouble understanding a few derivations in my book for least squares regression. $\textbf{Question 1}$: If $\textbf{M}^0 \textbf{i} = [\textbf{I} - \frac{1}{n}\textbf{ii'}]\...
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66 views

Find the number of elements in a quotient ring satisfying $a^2 = a$

Let $f(T)=T^3+T^2+2T+2 \in \mathbb{Z}[T]$ and let $I$ be the ideal of the ring $\mathbb{Z}[T]$ generated by $f(T)$ and $5$. Find the number of elements $z \in \mathbb{Z}[T]/I$ such that $z^2=z$. Some ...
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Is there a name for an associative algebraic structure in which everything is irreducible?

Let $A$ be a set and $\ast$ a binary operator on that set. Let us suppose that $(A,\ast)$ satisfies the following axioms: For all $x,y,z \in A$, $x \ast (y \ast z) = (x \ast y) \ast z$ For all $x,y,z ...
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Relation between simple roots and idempotents of quotient ring

Given a ring $A$ and a polynomial $p\in A[x]$, write $Z(p,A)$ for the simple roots in $A$ of $p\in A[x]$. On the other hand, consider the set $\mathrm{Idemp}(A[x]/(p))$ of idempotents of the quotient. ...
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1answer
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lifting orthogonal idempotents (induction step)

I'm trying to prove (by induction on $n$) that whenever $I$ is a nilpotent ideal of a ring $R$, and $r_1+I,\ldots,r_n+I$ form a complete set of orthogonal idempotents in the quotient $R/I$, there ...
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1answer
155 views

If $a^2=a$ find the inverse of $1+a$ [closed]

If $a^2=a$ find the inverse of $1+a$. Note that $a\in R$ where $R$ is a ring and inverse of $1+a$ exists in $R$ I tried $(1-a)(1+a)=1-a^2=1-a$ But I need $1$ in the RHS.How to get it?
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Proving that $\mathbf{(H-\frac{1}{n}J_n)}$ is indempotent

I am trying to show that the matrix $\mathbf{(H-\frac{1}{n}J_n)}$ is idempotent where $\mathbf{H}$ is the Hat-matrix (Projection matrix) of linear regression and $J_n$ is the $n\times n$ matrix with $...
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Idempotents and number of proper cyclic codes

So, there was a question on my exam last year as follows: Using idempotents, determine the number of proper cyclic codes of length 17. I got the question right, but I can't remember how to do the ...
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1answer
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GRE 9768 #60 1. Does $(s+t)^2=s^2+t^2$ imply $s+s=0$? 2. Idempotent matrices do not form a ring?

GRE 9768 #60 on what appears to be Boolean rings: Ian Coley's approach is to prove $(I)$ and $(I) \implies (II) \implies (III)$ I think $(II) \implies (I)$. My attempt: $$(s+t)^2=s^2+t^2 \...
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4answers
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Finding the idempotent matrix

I am faced with the problem: Let $p$ $=$ $\begin{pmatrix} 1\\2 \end{pmatrix}.$ Find the idempotent matrix M such that $Mv$ is orthogonal to $p$ for any $2 \times 1$ vector $v$. I understand that ...
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Question about the transpose of identity and hat matrices

So I know that because the hat matrix and identity matrix are both symmetric, $H^{T}=H$ and $I^{T}=I$, respectively so would $(I-H)^{T}=(I-H)$? Sorry if the answer is obvious, I am missing a step ...
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What property of a function describes $f\left(f\left(f\left(\phi, x\right), y\right), z\right) == f\left(\phi, z\right)$?

How would you describe a function that has the relationship $f\left(f\left(f\left(\phi, x\right), y\right), z\right) == f\left(\phi, z\right)$? given that $f\left(\phi, x\right)$ is idempotent (i.e., $...
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1answer
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How to dram Hasse Diagram in a lattice theory

Let $E$ be the set of all nonzero idempotents from the commutative ring $\mathcal{A}$. An atom in $E$ is an element $e$ in that set such that $ef = f$ for some $f \in E$ implies $f = e$. In other ...
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Peirce decomposition of a ring: must the ideal generators be idempotent in characteristic 2?

I'm considering this particular statement of the Peirce decomposition of a ring: If a commutative unital ring $R$ can be written as an internal direct sum of two of it's proper ideals $I$ and $J$, ...
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Understanding the linear transformation given by matrix multiplication

I am working on the following problem. I found it in an old qualifying exam, but I'm not sure of its original source. It asks: Let $A$ be an $n \times n$ matrix with entries in $\mathbb{R}$. Let $\...
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Transpositions and idempotents in matrix algebras

Let $A$ be a semi-simple algebra over $\mathbb{C}$. An idempotent $x$ in $A$ is an element of $A$ such that $x^2 = x$. A minimal non-zero idempotent $y$ of $A$ is a non-zero idempotent of $A$ such ...
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$A$ is a symmetric matrix such that $A^4=A$. Prove that $A$ is idempotent

Let $A$ be a real symmetric matrix such that $A^4=A$. Prove that $A$ is idempotent. I have tried using eigenvalues and only inferred that the eigen values may be $0,1$. But I cannot proceed with this....
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$R$ is artinian $\implies$ $(eR)_R$ is local $R$-module for an idempotent $e \in R$

Let $R$ be a ring with unity $1$ and artinian. Consider $R$ as right $R$-module $R_R$. Since it is artinian we have a finite decomposition of $R$ into right ideals: $$R_R = A_1 \oplus ... \oplus A_n$$...
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A characterization for idempotent lifting property

Let $I$ be an ideal in a commutative ring $R$ with $1$ and let $g+I$ be an idempotent element of $R/I$. We say that this idempotent can be lifted modulo $I$ in case there is an idempotent $e^2=e\in R$...
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Homotopy idempotent maps over a finite product of spheres

Let $X$ be a topological space. A continuous map $h:X\longrightarrow X$ is called homotopy idempotent if $h\circ h\simeq h$. My question is: What is the number of homotopy classes of homotopy ...
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Decomposition of idempotent matrix

Let be $P\in \mathbb{R}^{n\times n}$ of rank $r\leq n$ and idempotent, i.e. $PP=P$. I want to show that there exist matrices $A\in \mathbb{R}^{n\times r}$ and $B\in \mathbb{R}^{r\times n}$ such that $...
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Lifting idempotents modulo an ideal and its radical

The following result is a direct consequence of Proposition 27.1 of F. W. Anderson and K .R. Fuller's "Rings and categories of modules". But I cannot prove it. Is there any hint? Definition: Let $A$ ...
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Need help in proving $P$ is a projection.

A bounded linear operator $P:\mathbb H\rightarrow\mathbb H$ on a Hilbert space $\mathbb H$ is a projection iff $P$ is Self-adjoint and idempotent Proof:Initially,i started by assuming $P$ to be a ...
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Can I use Idempotency Here?

If I have statement like: AB + 'AC + BC, can I use Idepotency to remove BC and simplify, or does the AND between AB and 'AC rules this out? More specifically to simplify: AB+BC+C'A - Commutativty ...
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Primitive idempotents of Cl(1,3) over the complex numbers

Simply put, I need to find all primitive idempotents of the Clifford Algebra $Cl(1,3)$ over the complex numbers. I have found some general results but they're only applicable over the real numbers. I ...
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Idempotents and Factorization

In the ring $\mathbb{F}_2[x]$, let $f(x)$ be a reducible polynomial of degree $n$. If we happen to know that there is a non-trivial idempotent $g$ in $\mathbb{F}_2[x] / (f(x))$, then one of gcd($f,g$) ...
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Idempotent endomorphisms on $\mathbb{Z}\times \mathbb{Z}$ [closed]

Recall that a homomorphism $f:G\longrightarrow G$ is called idempotent if $f\circ f=f$. What are idempotent homomorphisms $f:\mathbb{Z}\times \mathbb{Z}\longrightarrow \mathbb{Z}\times \mathbb{Z}$ ...
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Do you know if eigenvectors of idempotent matrices are always orthonormal?

I'm currently studying Econometrics theory, and I'm stuck in my problem set where it is asked if "eigenvectors of idempotent matrices are always orthonormal". How can I justify that? Cheers.
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Regular Ring & principal ideals of idempotents [closed]

If $R$ be Regular Ring. And for any two idempotent elements $e$ & $f$ there exist a idempotent element $g$ such that $Re+Rf=Rg$
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Must an idempotent matrix be symmetric?

A matrix $A$ is idempotent if: $$AA = A$$ Is it true that all such matrices are symmetric?
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1answer
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Proving lemma about centrality of idempotent elements in a Ring with no nilpotent elements.

Reading the class notes I stumbled upon an unproved lemma that I am having issues proving. The lemma is, if $R$ is a ring with no non-zero nilpotent elements and $e$ is idempotent then $e$ is central....
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Idempotents over a ring with zero divisors

Let $R = \mathbb{Z}/4\mathbb{Z} = \{0, 1, 2, 3\}$ and the group $G = \mathbb{Z}/2\mathbb{Z} = \{e, a\}$. Consider the group ring $R[G]$. I have read somewhere (1) that $\frac{e + a}{2}$ and $\frac{e -...
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Is it true that $X(X'X)^{-1}X'-J/n$ is idempotent, where $J$ is an $n$ by $n$ matrix of ones?

$X$ is a full column rank $n$ by $p$ matrix with the first column a vector of ones. Now the I was trying to prove, from a different approach that the SSR/variance is Chi square but this means I have ...
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Is $A^3=A$ a condition for idempotency of matrices?

Given that $A$ and $B$ are two idempotent (square) matrices of same order, $AB+BA=AB-BA=O$ (Where $O$ is the null matrix of the same order). Prove that both $A+B$ and $A-B$ are idempotent. I ...
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Composition series of regular module

Assuming I know all (primitive) idempotents of an algebra $A$, and its centre, is there a way to obtain a composition series for the regular module $_A A$, and the simple modules of $A$?
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Relations of structures related to conjugate idempotents

If $e$ and $f$ are conjugate idempotents in some algebra $A$, I guess the modules $Ae$ and $Af$ should be isomorphic, as well as the algebras $eAe$ and $fAf$ . Are the maps canonically given by just ...
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Abstract Algebra: Describe the idempotents in $(R/I, R/J)$.

Suppose $R$ is a ring with ideals $I,J$ such that $I+J=R$ and $I\cap J=0$. I have proved that, if $\phi,\psi$ are the natural homomorphisms associated with $I,J$ respectively, then $\chi(a)=(\phi(a),\...
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Idempotent substitution $\theta$

Exercise: A substitution $\theta = \{x_1\leftarrow t_1, \dots, x_n\leftarrow t_n\}$ is idempotent iff $\theta = \theta\theta$. Let $V$ be the set of variables occurring in the terms $\{t_1, \...
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Prove that $A-I_n$ is idempotent

Let $A ∈ R^{n×n}$ be a nonzero singular matrix with the property that $A^2 = A$. Show that $A − I_n$ is also idempotent. I tried the following: $(A - I_n)^2 = (A-I_n)(A-I_n) = A(A-I_n)-I_n(A-...
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The idempotent elements of Eisenstein Integers

Let a+bω be an Eisenstein integer. An idempotent element of $ \mathbb Z_n[\omega]$ is $(a+b\omega)^2 \equiv (a+b\omega)\pmod{n} $, where $\omega^2=-\omega-1$ But it follows that the idempotent element ...
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1answer
63 views

What is the set {$e\in(R/ I)\times(R/J): e$ is idempotent}

What is the set {$e\in(R/ I)\times(R/J): e$ is idempotent}? I am trying to solve part (d) of Chaper 11 problem 6.8 from Artin's Algebra textbook. I have already solved a,b, and c. Let $I$ and $J$ ...
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206 views

Prove that an idempotent element must be either 0, 1 or a zero-divisor.

This is what I've come up with for the proof, but I feel like I'm missing a huge piece of the puzzle here. Any thoughts? Proof. Suppose $R$ is an integral domain and let $a\in R$ be any idempotent. ...
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Generalized Idempotency of Multi-variable Function

I am wondering what a generalization of idempotency would look like for a function of multiple variables. For instance, for basic idempotency of a function of one variable, we have: ...