# Tagged Questions

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

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### Minimal ideal in a ring which is generated by an idempotent element.

Let $R$ be a commutative ring with unity and $M$ be a minimal ideal of $R$ such that $M = Re$ where $e$ is an idempotent element in $R$. Then $R = Re \oplus R(1-e)$ I am not able to see, in order ...
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### Idempotents in commutative ring of characteristic 2 form a subring

Question: In a commutative ring of characteristic $2$, want to show that the idempotents form a subring. Subring Test is probably the way to go. It is easy to verify the identity element in ...
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### 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 $(\implies)$...
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### 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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### 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 ...
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### 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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### 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. [closed]

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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### 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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### 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 ...
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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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