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- 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 figures "
- 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 Operators B
- Proof of $(\\mathbb{Z} m\\mathbb{Z}) \\otimes_\\mathbb{Z} (\\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
- $\\operatorname{Hom}_{G}(V,W) \\cong \\operatorname{Hom}{G}(\\mathbf{1 . . .
Let $G$ be a group with some completely reducible finite-dimensional representations $V,W$, and let $\\mathbf{1}$ be the trivial irreducible representation I'm
- $A I \\otimes_A A J \\cong A (I+J)$ - Mathematics Stack Exchange
I'm assuming you mean this isomorphism as an isomorphism of A -algebras (I'm going to assume A is commutative so this makes sense) Then A I has the following universal property: for any A -algebra B such that the elements of I map to zero under the structural map A → B, there is a unique A -algebra map A I → B Similarly for A J The tensor product (A I) ⊗ A (A J) is the
- abstract algebra - If $\mathbb Q \otimes_\mathbb Z \mathbb Q \cong . . .
In Dummit Foote, it is an exercise to show that $\mathbb Q \otimes_\mathbb Z \mathbb Q$ is a $1$-dimensional $\mathbb Q$-vector space This is fairly easy: a $\mathbb Q$-basis for $\mathbb Q \
- Let - Mathematics Stack Exchange
Let M M and N N be two normal subgroups of G G Show that M ∩ N M ∩ N is also normal in G G Furthermore,if G = MN G = M N then show that
- $\\Bbb Z[i] (a+bi)\\cong \\Bbb Z (a^2+b^2)$ if $(a,b)=1$. Gaussian . . .
A very basic ring theory question, which I am not able to solve How does one show that Z[i] (3 − i) ≅Z 10Z Z [i] (3 − i) ≅ Z 10 Z Extending the result: Z[i] (a − ib) ≅Z (a2 +b2)Z Z [i] (a − i b) ≅ Z (a 2 + b 2) Z, if a, b a, b are relatively prime My attempt was to define a map, φ: Z[i] → Z 10Z φ: Z [i] → Z 10 Z and show that the kernel is the ideal generated
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