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

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1answer
49 views

If $e$ and $f$ are idempotent and $e(e+f)f=e+f$, then prove that $e=f$

Let $R$ be an associative ring and let the idempotents $e$ and $f$ belongs to $R$. Then prove that $$e(e+f)f=e+f \iff e=f$$ The only if part is very easy and I have proved what about the $(\implies)$...
3
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0answers
24 views

Proving Logical equivalence [5-26]

I have to prove a problem statement with logical equivalences but I seem to keep getting stuck. Here is the problem: $$ [(q \to p) \land \lnot p] \to (p \land q) \equiv p \lor q $$ Here is the work I ...
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1answer
15 views

Proving Logic statement

So I have an statement that I need to prove using Logical Equivalences: $$(p\land q) \lor [p \land (\lnot( \lnot p \lor q)) ] \equiv p $$ I made it through some steps but I can't seem to make it to ...
1
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1answer
16 views

Relationship between idempotents in semisimple ring.

Let R be a semisimple ring with identity. If $e$ and $e'$ are idempotents in R such that $Re \simeq Re'$ then there exists $a \in R^\times$ such that $e' = aea^{-1}$ Attempt I know from a previous ...
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0answers
81 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 ...
6
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3answers
121 views

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
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1answer
98 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?
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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
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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 ...
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1answer
94 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 $[x_m,\dotsc ,x_1]^T$. Is $...
0
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3answers
70 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
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0answers
55 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 decompositions $V=X\dotplus Y$. Attempt: We have $e^2 = e$. Therefore, we can write ...
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2answers
111 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 is ...
3
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0answers
87 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 $\...
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1answer
25 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 +...
2
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2answers
43 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?
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3answers
64 views

Studying the intersection $(X)\cap (X^{2}-Y+1)\subseteq\mathbb{R}[X,Y]$.

I am trying to find the intersection of ideals $$ (X)\cap (X^{2}-Y+1)\subseteq\mathbb{R}[X,Y]. $$ This is what I have tried: $$ f\in(X^{2}-Y+1)\Rightarrow f=g\cdot (X^{2}-Y+1)\text{ for certain }g\...
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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 ...
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3answers
36 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
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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
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1answer
23 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
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1answer
20 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
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2answers
54 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 $A$, $e \neq 0,1$,...
4
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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
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3answers
38 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
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1answer
44 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 ...
6
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1answer
113 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 ...
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0answers
51 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 the ...
3
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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
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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 ...
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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 $A\...
0
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1answer
48 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 ...
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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
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0answers
40 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
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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
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1answer
37 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 ...
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4answers
480 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 \\ \end{...
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2answers
73 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 $...
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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 ...
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2answers
54 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
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1answer
323 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 know ...
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4answers
248 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.
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2answers
60 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 n}:...
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1answer
129 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
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1answer
66 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
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4answers
419 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 have $...
3
votes
1answer
147 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
48 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 $S$ is ...
0
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1answer
46 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 R(1-e)...