List questions from ring-isomorphism

Published on 06 May 2021 - 19:15

62

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$\Bbb{R}[x]/\langle(x+1)(x^2+1)\rangle$ is isomorphic to $\Bbb{R}\times\Bbb{R}\times\Bbb{R}$

Published on 07 May 2021 - 15:53

156

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$\mathbb{R}^3$ isn't isomorphic to $\mathbb{R} \times \mathbb{C}$

Published on 08 May 2021 - 13:16

44

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Show that $\phi$ is an isomorphism in this double quotient

Published on 10 May 2021 - 13:45

186

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If R is an integral domain then (R,+) is NOT isomorphic to (U(R),*)

Published on 11 May 2021 - 6:51

69

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Let $\phi : \mathbb Q(\sqrt2)\to \mathbb Q(\sqrt2)$ be an isomorphism. Show that if $\phi(1) = 1$, then $\phi(a) = a$ for all $a \in \mathbb Q$

Published on 14 May 2021 - 18:03

62

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How to construct a ring isomorphism in this case?

Published on 19 May 2021 - 18:05

162

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Constructing a ring isomorphism between $\mathbb{Z}_7$ and $2\mathbb{Z}_{14}$

Published on 02 Jun 2021 - 19:18

198

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Showing that $\mathbb{R}[X,Y]/(X^2+1, Y^2+1) \cong \mathbb{C} \times \mathbb{C}$

Published on 05 Jun 2021 - 14:28

74

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Ring isomorphism between two quotient rings

Published on 10 Jun 2021 - 1:36

69

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First isomorphism theorem with sequences

Published on 14 Jun 2021 - 0:38

75

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there's a surjective homomorphism between $\mathbb{R}[x]$ and $\begin{equation}\begin{pmatrix} a & b \\-3b & a\end{pmatrix}\end{equation}$ $a,b$ reals

Published on 20 Jun 2021 - 19:02

58

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Subring of $\mathbb{Z} \times \mathbb{Z}$

Published on 24 Jun 2021 - 0:57

156

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Show isomorphism between localization of rings

Published on 29 Jun 2021 - 22:35

76

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$\mathbb{R}[x,y]/(x^3,y) \cong \mathbb{R}[x,y]/(x^2,y^2)$

Published on 05 Jul 2021 - 11:11

List Question

When $R[t]/f(t) \cong R[[t]]/f(t)$

$\Bbb{R}[x]/\langle(x+1)(x^2+1)\rangle$ is isomorphic to $\Bbb{R}\times\Bbb{R}\times\Bbb{R}$

$\mathbb{R}^3$ isn't isomorphic to $\mathbb{R} \times \mathbb{C}$

Show that $\phi$ is an isomorphism in this double quotient

If R is an integral domain then (R,+) is NOT isomorphic to (U(R),*)

Let $\phi : \mathbb Q(\sqrt2)\to \mathbb Q(\sqrt2)$ be an isomorphism. Show that if $\phi(1) = 1$, then $\phi(a) = a$ for all $a \in \mathbb Q$

How to construct a ring isomorphism in this case?

Constructing a ring isomorphism between $\mathbb{Z}_7$ and $2\mathbb{Z}_{14}$

Showing that $\mathbb{R}[X,Y]/(X^2+1, Y^2+1) \cong \mathbb{C} \times \mathbb{C}$

Ring isomorphism between two quotient rings

First isomorphism theorem with sequences

there's a surjective homomorphism between $\mathbb{R}[x]$ and $\begin{equation}\begin{pmatrix} a & b \\-3b & a\end{pmatrix}\end{equation}$ $a,b$ reals

Subring of $\mathbb{Z} \times \mathbb{Z}$

Show isomorphism between localization of rings

$\mathbb{R}[x,y]/(x^3,y) \cong \mathbb{R}[x,y]/(x^2,y^2)$

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