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

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7
votes
0answers
64 views

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 ...
3
votes
0answers
25 views
+50

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 ...
9
votes
1answer
81 views

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?
0
votes
0answers
24 views

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 ...
0
votes
2answers
24 views

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 ...
3
votes
1answer
67 views

Question about idempotent matrices. [closed]

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 $[x_m,\dotsc ,x_1]^T$. ...
0
votes
3answers
62 views

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 ...
1
vote
0answers
47 views

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 ...
5
votes
2answers
92 views

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 ...
3
votes
0answers
86 views

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 ...
2
votes
1answer
24 views

if $e$ is an idempotent, then $Re$ is a projective module

I would like to show that if $R$ is a ring with $1$, and $e$ is an idempotent in $R$, then $Re$ is a projective module. My idea is to show that $R = Re \oplus R(1-e)$. It's clear to me that $R = Re ...
1
vote
2answers
35 views

commutative ring which have every maximal ideal generated by an idempotent [closed]

Can you help me with one example of commutative ring which have every maximal ideal generated by an idempotent?
0
votes
1answer
20 views

On the equivalence condition of p.p. rings

A ring is known to be left p.p. if every principal left ideal is projective. It is well known that this condition is equivalent to the fact that every annihilator of each element is generated by an ...
-1
votes
3answers
34 views

How can I prove the fact that “a prime ring has no nontrivial central idempotent”?

It seems like "a prime ring has no nontrivial central idempotent" is a well known fact since the Book of Rowen and Goodearl and lots of other papers use this fact freely. However, I cannot find the ...
0
votes
3answers
64 views

Even-odd multiplicative algebraic structure with idempotency? [closed]

What is the algebraic structure for the multiplications of even elements and odd elements? Please notice that $o*o=o$, $e*e=e$ (idempotency) and $o\not = e$. 1st structure is such that even times ...
0
votes
1answer
18 views

Idempotents with arbitrarily large norms in Banach algebras

While nonzero projections in $C^*$-algebras have norm 1, there is no such restriction for idempotents in Banach algebras. What is an example of a Banach algebra that has idempotents of arbitrarily ...
1
vote
1answer
16 views

Why is the Young symmetrizer non-zero?

Suppose $\lambda$ is a partition of the natural number $n$ and $T$ is a standard Young Tableaux of shape $\lambda$. Let $$P_{\lambda}:=\lbrace g\in S_n:g\text{ preserves the rows of }T\rbrace$$ and ...
0
votes
2answers
52 views

Prove that the sum of ideals of a ring A equals A and its intersection is zero.

I've been looking at a couple of ring theory exercises and there's this one I don't know how to do it. It goes like this. $A$ is a commutative unital ring, and $e$ an element of $A$, $e \neq ...
4
votes
0answers
45 views

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 ...
0
votes
3answers
37 views

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: ...
0
votes
1answer
35 views

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 ...
5
votes
1answer
111 views

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 ...
1
vote
0answers
50 views

nil Jacobson radicals

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 ...
3
votes
0answers
44 views

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 ...
3
votes
2answers
38 views

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 ...
2
votes
2answers
78 views

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 ...
0
votes
1answer
47 views

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 ...
1
vote
2answers
32 views

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 ...
0
votes
0answers
39 views

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) ...
0
votes
2answers
23 views

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 ...
0
votes
1answer
36 views

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 ...
1
vote
4answers
361 views

Proving that if $AB=A$ and $BA=B$, then both matrices are idempotent

Let $A, B$ be two matrices such that $AB=A$ and $BA=B$, how do I show that $A\cdot A=A$ and $B\cdot B=B$? Steps I took: Let $A= \left[\begin{array}{rr} a & b \\ c & d \\ ...
4
votes
2answers
71 views

Does inverse image of idempotent element contain an idempotent element? [closed]

Suppose $f\colon \oplus A_i\to\oplus B_j$ is a ring homomorphism, where $A_i,B_j$ are local rings. Both sides have finitely many summands. Suppose an idempotent $x$ (i.e. $x=x^2$) is in the image of ...
0
votes
2answers
75 views

Let $A ∈ M_{n×n}(\mathbb{R})$ differ from $I$ and $O$. If $A$ is idempotent, show that its Jordan canonical form is a diagonal matrix.

Let $A ∈ M_{n×n}(\mathbb{R})$ differ from $I$ and $O$. If $A$ is idempotent, show that its Jordan canonical form is a diagonal matrix. I'm not sure how to do this. Any solutions/hints are greatly ...
1
vote
2answers
52 views

Finding (orthogonal) idempotents in a quotient of a polynomial ring

I wish to identify all of the non-trivial idempotents in the ring $$ \Bbbk[x]/\langle x(x+1)^2(x+2)^2(x+3)\rangle.$$ I have reason to suspect that there are at least four which are, in addition, ...
0
votes
1answer
311 views

$\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$ for $A$ idempotent matrix

Let $A$ be a square matrix of order $n$. Prove that if $A^2=A$ then $\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$. I tried to bring the $A$ over to the left hand side and factorise it out, but do not ...
2
votes
4answers
228 views

Let A be a square matrix of order n. Prove that if $A^2 = A$, then $\operatorname{rank}(A) + \operatorname{rank}(I - A) = n$. [duplicate]

$A(I-A) = 0\implies\operatorname{rank}(A) + \operatorname{rank}(I-A)\le n$. I managed to get this but wasn't able to go further. Any help would be appreciated.
0
votes
2answers
58 views

Convex hull of idempotent matrices

What is the convex hull of the set of $n\times n$ (potentially asymmetric) idempotent matrices? Apparently the powers that be want more information: Consider the set $S:=\{A\in\mathbb{R}^{n\times ...
1
vote
1answer
114 views

Idempotent projection operators in a Hilbert space

Let $H$ be a Hilbert-space and $S$ be a sub(Hilbert)space such that: $$H = S \oplus S^\perp$$ Then the projection operators are defined as: $$P_S: H\to S; x = u + v \mapsto u \quad\quad P_{S\perp}: ...
1
vote
1answer
65 views

The $e$ in the operator $exe'$

I am learning the group algebra. A way to study the irreducible representations of a finite group is to use the idempotent operators in the group algebra. Suppose $e$ and $e'$ are the primitive ...
14
votes
4answers
406 views

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 ...
3
votes
1answer
129 views

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 ...
1
vote
2answers
46 views

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 $S$ ...
0
votes
1answer
43 views

Show that $R\cong Re\times R(1-e)$

An element $e\in R$ is called idempotent if $e^2=e$. Assume that $e$ is idempotent and $er=re$ $\forall r\in R$. Prove that $Re$ and $R(1-e)$ are two sided ideals of $R$ and $R\cong Re\times ...
3
votes
1answer
74 views

If $R$ is a simple ring, is every corner $eRe$ simple?

Assume that $R$ is a ring, not necessarily commutative nor unital. Let $e$ be an idempotent in $R$. Is there a correspondence between ideals of $R$ and ideals of the corner ring $eRe = \{ere : r\in ...
5
votes
1answer
55 views

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 ...
2
votes
0answers
97 views

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 $a,x,y\in R$, ...
1
vote
0answers
35 views

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 ...
1
vote
0answers
38 views

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$ ...
1
vote
1answer
23 views

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 ...