Intereting Posts

Any more cyclic quintics?
isolated non-normal surface singularity
Why is recursion theory suffering from terminological bloat?
Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?
Given the Cauchy's problem: $y'' = 1, y(0) = 0, y'(0) = 0$. Why finite difference method doesn't agree with recurrence equation?
Representation of symmetric functions
How to show $e^{2 \pi i \theta}$ is not algebraic.
If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$?
Showing a subset of the torus is dense
If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$
Riemann-integrable (improperly) but not Lebesgue-integrable
Why is Klein's quartic curve not hyperelliptic
Enumerations of the rationals with summable gaps $(q_i-q_{i-1})^2$
Show that $x^2 + x + 12 = 3y^5$ has no integer solutions.
How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$

Let $V$ be a finite-dimensional vector space over field $k$ and $R = \text{End}_k V$. How do I see that any left ideal of $R$ takes on the form $Rr$ for some suitable element $r \in R$?

- Showing the sum of a C* subalgebra and ideal is itself a C* subalgebra
- Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)
- 2 Tricks to prove Every group with an identity and x*x = identity is Abelian - Fraleigh p. 48 4.32
- Unique factorization domain that is not a Principal ideal domain
- Cancellation Law for External direct product
- Why are fields with characteristic 2 so pathological?
- A problem with tensor products
- What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?
- Trace and Norm of a separable extension.
- Show that $(2+i)$ is a prime ideal

Since $R$ is a semisimple ring, every left ideal will split out like this: $R=L\oplus L’$. In this decomposition, $1=e+e’$ where $e\in L$ and $e’\in L’$.

It’s an exercise to prove that $e^2=e$ and $(e’)^2=e’$. (Really, $e’=1-e$.)

It’s another exercise to show that $L=Re$. Just remember that the sum is direct.

- Problem on Linear Diophantine Equation over 3 variables
- Proof of the Banachâ€“Alaoglu theorem
- Sum from 0 to n of $ n \choose i $?
- How does $\cos x=\frac12(e^{ix}+e^{-ix})$?
- A decomposition of a differentiable function
- simple tools to extract Re,Im,Abs… of any complex function
- Boundedness of solutions for the Laplacian
- Is there a unique finitely-additive extension of the length function to all real subsets?
- Deriving the addition formula of $\sin u$ from a total differential equation
- No solvable subgroups of $\operatorname{SL}_2(\mathbf Z)$ of finite index
- Constructive Proof of Kronecker-Weber?
- independent, identically distributed (IID) random variables
- Lebesgue integration by substitution
- Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$
- Do sets whose power sets have the same cardinality, have the same cardinality?