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Mar
14
revised $GL(n, R)$ is abelian then $n = 1$
added 4 characters in body
Mar
13
comment Show an ideal is a finitely generated projective module via a split exact sequence
$K$ is the kernel of $\phi$.
Mar
13
comment Show an ideal is a finitely generated projective module via a split exact sequence
Yes, it means $R\oplus R\oplus\cdots\oplus R$, $m$ times.
Mar
13
reviewed No Action Needed equivalent ways of writing the spectral norm
Mar
13
reviewed Leave Open A relationship among the first $n+1$ primes
Mar
13
answered Show an ideal is a finitely generated projective module via a split exact sequence
Mar
13
comment Show an ideal is a finitely generated projective module via a split exact sequence
Have you tried something along this line: $1=\alpha_1(i_1)+\cdots+\alpha_m(i_m)$, and see whether the morphism $\oplus^mR\to I, r_1\oplus\cdots\oplus r_m\mapsto\sum_{k=1}^m r_ki_k$ is onto and split?
Mar
12
comment Is there a combinatorial explanation for the identity: $\sum_{k=1}^n c(n,k)2^k = (n+1)!$
Do you know the identity $$X(X+1)\cdots(X+n-1)=\sum_{k=1}^n\begin{bmatrix}n\\k\end{bmatrix}x^k\;?$$
Mar
12
comment General Form of Orthogonal Upper Triangular Matrices
Use the definition of being an orthogonal matrix: the columns (say) form an orthonormal basis. The first column looks like so $$\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix}$$ and this forces all the other coefficients in the first row to be zero. Hence the second column must be $$\begin{pmatrix}0\\1\\0\\\vdots\\0\end{pmatrix}$$ This pattern continues, and shows that your matrix was the identity matrix.
Mar
12
reviewed Close Number of solutions of arithmetic funtion's equation.
Mar
12
reviewed Close This relation exists but how to show this?
Mar
12
comment If $a^{2k}=a$ for all elements of a ring then $a= -a$ for all $a$
Hint: $(-1)^{2k}=1$ for all $k$.
Mar
8
answered $d+1$ distinct points of a rational normal curve in $\mathbb{P}^{d}$ are linearly independent
Mar
8
revised Why is $\deg(f)$ well-defined?
edited body
Mar
8
answered Why is $\deg(f)$ well-defined?
Mar
8
revised Infinite dimensional space of solutions
added 441 characters in body
Mar
8
answered Infinite dimensional space of solutions
Mar
6
comment Show that $ℤ[\sqrt{8}][x]$ is not a UFD
I'm not sure this precise example will work, but you could try to invalidate the unique factorization by considering something like $8=(4+\sqrt{8})(4-\sqrt{8})=\sqrt{8}\cdot\sqrt{8}=2\cdot 2\cdot 2$.
Mar
4
comment Fibers in Covering Spaces
Locally finite or locally constant?
Mar
4
comment Topology and Smooth Structure on the Bundle of Covariant $k$-Tensors
Possibly called the "smooth bundle construction lemma"?