Tagged Questions

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

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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 ...
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Identity of a ring as two different sums of idempotents

Let $R$ be any ring with identity $1_R$. Prove that if there exist idempotents $e_1,..., e_n, e'_1,...,e'_n \in R$ such that $$1_R = e_1 +...+ e_n = e'_1+... e'_n$$ then the following conditions ...
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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?
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Composition series of a regular module.

Suppose $A$ is an $k$-algebra with basis ${1,e,s,t}$ and multiplication is given by $$e^2 = e, es = s, te = t, s^2=t^2=se=et=st=ts=0.$$ I am trying to find the composition series for ...
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Show that Jordan block is idempotent if it is 0 or 1

An idempotent matrix $M$ is one such that $M^2 = M$. A Jordan block has its eigenvalue $\lambda$ on its diagonal and 1 on the superdiagonal. I figure that in order to ensure that $M^2=M$, it makes ...
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Let $E$ be the $m \times m$ matrix that extracts the "even part" of an $m$-vector $Ex = (x+Fx)/2$, where $F$ is the $m\times m$ matrix that flips $[x_1,\dotsc ,x_m]^{T}$ to $[x_m,\dotsc ,x_1]^T$. ...
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Each prime ideal contains an idempotent element

An element of ring $e$ is called idempotent iff $e^2=e$. Let $R$ be a commutative ring that contains the identity element and a non-trivial idempotent element. I want to show that each of ...
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Idempotent endomorphisms generate a direct sum decomposition

Show that there is a one-to-one correspondence between indempotents $e\in\operatorname{End}_R(V)$ and direct sum decomposition $V=X\dotplus Y$. Attempt: We have $e^2 = e$. Therefore, we can write an ...
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Idempotent ideals in certain commutative rings

Let $R$ be a commutative ring with zero Jacobson radical such that each maximal ideal of $R$ is idempotent. Does it guarantee that each ideal is idempotent? I know only that if each maximal ideal ...
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Eigenspace of convex combination of two idempotent matrices

Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix $$H_\mu:=\mu H_1+(1-\mu)H_2.$$ I'm looking for a description of ...
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Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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Show that if $\rho$ is idempotent then $\rho$ acts as the identity on $\rho(V)$

A linear map $V \xrightarrow{\rho} V$ is idempotent if $\rho\rho = \rho$. Show that if $\rho$ is idempotent then $\rho$ acts as the identity on $\rho(V)$. (Such linear maps are called projections: ...
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eigenvalues of composite matrix

I have a $n$ x $n$ symmetric positive definite matrix $C$, and a symmetric idempotent matrix $M$, given by $I-\frac{1}{n}J_n$ where $J_n$ is a $n$ x $n$ matrix of all ones. Additionally, all the ...
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K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
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Let $f$ be an idempotent element in a ring $S$ with Jacobson radical $J$ so that both $fJf$ and $(1-f)J(1-f)$ are nil. I guess that $J$ is nil too, but I am not sure. I know that the former is ...
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Homomorphisms of modules over a corner ring

Let $R$ be a Noetherian ring and suppose that we can write $1 = e_1 + e_2 + \dots + e_n$ where the $e_i$ are pairwise orthogonal idempotents. Let $S = e_1 S e_1$, and consider the right $S$-modules ...
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Idempotent $17\times 17$ Matrix Jordan Normal Form

I have a $17\times 17$ idempotent matrix $M$ with rank $10$ $M$ is idempotent so $M^2=M \rightarrow M(M-1)=0$. So the minimum polynomial is $x(x-1)$. The eigenvalues of $M$ are $x=0$ and $x=1$. The ...
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If $A,B,A+I,A+B$ are idempotent matrices how to show that $AB=BA$?

If $A,B,A+I,A+B$ are idempotent matrices how to show that $AB=BA$ ? MY ATTEMPT: $A\cdot A=A$ $B\cdot B=B$ $(A+I)\cdot (A+I)=A+I$ or, $A\cdot A+A\cdot I+I\cdot A+I\cdot I=A+I$ which implies ...
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Idempotent ideals having only idempotents

I search for sufficient conditions for a ring $R$ so that any idempotent ideal constitutes only of idempotent elements of $R$. Of course, in the commutative case, any finitely generated idempotent ...
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Nonminimal idempotents

Let $S$ be a semigroup with operation $+$. Note that we do not assume $S$ is abelian. We say that $I\subset S$ is a left ideal if and only if for all $s\in S$, $s+I\subseteq I$, and we say that an ...
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S an inverse semigroup with semilattice of idempotents E and $\sigma$ the minimum group congruence on S.

Let S be an inverse semigroup with semilattice of idempotents E, and let $\sigma$ be the minimum group congruence on S. Show that the following statements are equivalent: (a) $x\sigma y$; (b) ...
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Find scalars that result in an idempotent matrix

For what scalars, $b$ and $c$ will the following matrix be idempotent: $bI_m + c1_m1_m^T$ I know that idempotency implies that each eigenvalue is 0 or 1, but I am still having trouble finding the b ...
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Prove $a,b\in R$ are idempotent iff $ab=0$.

Let $R$ be a ring. An element $a\in R$ is called idempotent if $a^2=a$. Suppose that R is a ring with unity $1\neq 0$, and that there are $a,b\in R$ such that $a+b=1$. Prove that $a,b$ are idempotent ...
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Idempotents in commutative ring

I have proven for $e$ an idempotent in a commutative ring $R$ that there exists an isomorphism $\phi:R\rightarrow R/eR\times R/(1-e)R$ given by $r\mapsto(r,r-re)$. Now I would like to prove that ...
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Idempotent in a ring of continuous functions?

Let $S$ be a set of continuous functions defined on an interval $[0,1]$. The addition and multiplication between two elements is defined as $(f+g)(x)=f(x)+g(x)$ and $(f\cdot g)(x)=f(x)g(x)$. So $S$ ...
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Question concerning a minimal faithful left ideal in an artin algebra

Let $B$ be an artin algebra an suppose there is a faithful projective-injective left $B$-module. Moreover, there is a minimal faithful left ideal $Be$ for some idempotent $e\in B$. 1) What does ...
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Idempotent semiring

Let $R$ be a semiring. For $a\in R$,we define $t_a(x)=x+a$ for all $x\in R$. Prove that $R$ is idempotent(with +) and $1$ has an infinite order if and only if for all $a,x,y\in R$, ...
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Ellis semigroup

I have only seen the construction of the Ellis semigroup applied to a group endowed with the discrete topology. Does the same idea works for any topological group? If not, where does it go wrong? (I ...
Determine all ring homomorphisms from $\Bbb Z$ $\oplus$ $\Bbb Z$ to $\Bbb Z$. [duplicate]
I got $(a,b) \to a$, $(a,b) \to b$ and $(a,b) \to 0$ these mappings to be homomorphisms just by hit and trial. So when I looked for it's solution, these were the ONLY homomorphisms from $\Bbb Z$ ...
Find Matrix $S$ such that $S'CS=\left(\begin{array}{cc} 1_{N-1} & 0 \\ 0 & 0 \end{array} \right)$ where $C:=1_N-\iota \iota'$
The Centering Matrix $C:=1_N-\iota \iota'$ has eigenvalue $1$ of multiplicity $n − 1$ and eigenvalue $0$ of multiplicity $1$. Therefore a matrix $S$ with columns consisting of eigenvectors of $C$ can ...