What does "$\cong$" sign represent? - Mathematics Stack Exchange
I came across this sign when reading some papers. I looked up Wikipedia. It says "The symbol "$\\cong$" is often used to indicate isomorphic algebraic structures or congruent geometric …
Difference between "≈", "≃", and "≅" - Mathematics Stack Exchange
In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"? The Unicode standard lists all of them inside the Mathematical …
Proof of $(\\mathbb{Z}/m\\mathbb{Z}) \\otimes_\\mathbb{Z} …
I've just started to learn about the tensor product and I want to show: $$ (\mathbb {Z}/m\mathbb {Z}) \otimes_\mathbb {Z} (\mathbb {Z} / n \mathbb {Z}) \cong \mathbb ...
A nonsplit short exact sequence of abelian groups with $B \\cong …
Jul 1, 2016 · The original sequence splits if and only if this sequence is exact on the right. If A A, B B and C C are of finite length as modules, this follows immediately just by counting lengths. …
abstract algebra - Prove that $\mathbb Z_ {m}\times\mathbb Z_ …
Prove that Zm ×Zn ≅Zmn Z m × Z n ≅ Z m n implies gcd(m, n) = 1 gcd (m, n) = 1. This is the converse of the Chinese remainder theorem in abstract algebra. Any help would be …
Computing the Canonical bundle $K_{\\mathbb{P}^n} \\cong …
Aug 22, 2023 · Here the det det means the highest exterior power. Then I don't know how can we extract the desired formula KPn ≅ O(−n − 1) K P n ≅ O (− n − 1) from this relation. I think that …
modules - How to prove that $R/I \otimes_R M \cong M / IM ...
Possible Duplicate: Showing that if R R is local and M M an R R -module, then M⊗R (R/m) ≅ M/mM M ⊗ R (R / m) ≅ M / m M. In one of the answers to one of my previous questions the …
Sum of 1 + 1/2 + 1/3 +.... + 1/n - Mathematics Stack Exchange
How do I calculate this sum in terms of 'n'? I know this is a harmonic progression, but I can't find how to calculate the summation of it. Also, is it an expansion of any mathematical function? 1 ...
Connected sum of projective plane $\\cong$ Klein bottle
Nov 28, 2014 · How can I see that the connected sum P2#P2 P 2 # P 2 of the projective plane is homeomorphic to the Klein bottle? I'm not necessarily looking for an explicit homeomorphism, …
Formula for $1^2+2^2+3^2+...+n^2$ - Mathematics Stack …
(n + 1)3 −n3 = 3n2 + 3n + 1 (n + 1) 3 − n 3 = 3 n 2 + 3 n + 1 - so it is clear that the n2 n 2 terms can be added (with some lower-order terms attached) by adding the differences of cubes, …