Intereting Posts

Characteristic function of Normal random variable squared
Show that the $\mathbb{Z}$-span $\mathbb{Z}b'_1+\cdots+ \mathbb{Z}b'_d$ of $B^+$ does not depend on the choice of $B$
for two positive numbers $a_1 < b_1$ define recursively the sequence $a_{n+1} = \sqrt{a_nb_n}$
The characteristic and minimal polynomial of a companion matrix
Showing $f(0) = 0$ and $|f'(x)| \leq M$ implies $|f(x)| \leq M |x|$.
Starting with $\frac{-1}{1}=\frac{1}{-1}$ and taking square root: proves $1=-1$
Calculate the binomial sum $ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i} $
Describing a Wave
Question about the totient function and congruence classes
Prove that if $f$ is bounded and nondecreasing on $(a,b)$ then lim $f(x) $as $x$ approaches $b$ from the left exists.
3 balls drawn from 1 urn – probability all same color (with/without replacement)
Find all $f(x)$ such that $f(gcd(x,y))=gcd(f(x),f(y))$
How to find $\lfloor 1/\sqrt{1}+1/\sqrt{2}+\dots+1/\sqrt{100}\rfloor $ without a calculator?
Permutations and Derangements
Prove Parseval Identity for $f \in C(\Bbb T) 2\pi$ periodic continuous functions

Let $R$ be a ring with unity (not necessarily commutative) and $I$ an ideal of $R$.

Suppose that for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$.

Note that $c$ is related to $a$. Now we have the following question:

Can we say that for every element $a\in I$ there exists an **idempotent** element $c\in I$ such that $ac=a$?

- Coker of powers of an endomorphism
- Compute the Jacobson radical of the group ring $\mathbb{F}_2S_3$.
- Every right principal ideal non-emptily intersects the center — what is that?
- Why are Dedekind-finite rings called so?
- Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?
- Dimension of division rings extension

Of course we have many examples such that the answer is true for them but in general we don’t know.

- Proving ring $R$ with unity is commutative if $(xy)^2 = x^2y^2$
- Vandermonde identity in a ring
- Order of $\mathbb{Z}/(1+i)$
- Prove that an infinite ring with finite quotient rings is an integral domain
- $R/M$ is a division ring
- Why only two binary operations?
- Correspondence theorem for rings.
- Lack of unique factorization of ideals
- Is a finite commutative ring with no zero-divisors always equal to the ideal generated by any of its nonzero elements
- Kaplansky's theorem of infinitely many right inverses in monoids?

Consider the ring $R$ of continuous functions $\mathbb R\to\mathbb R$ with compact support.

There are no non-zero idempotents in this ring, yet your condition holds. Indeed, if $a\in R$, let $c\in R$ be any function which is equal to $1$ on the support of $a$.

**Later** This ring does not have a unit, and you wwanted it to have one. But if $R$ does have a unit then your question is trivial: you can always take $c=1$!

- Liouville function and perfect square
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- Evaluate $\sum_{n=1}^\infty \frac{n}{2^n}$.
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- Counting Components via Spectra of Adjacency Matrices
- Understanding dot product of continuous functions
- How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+…+a_n}{n}=A$
- For integers $a\ge b\ge 2$, is $f(a,b) = a^b + b^a$ injective?