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Questions tagged [ring-isomorphism]

This tag is for questions regarding to Ring Isomorphisms, a ring homomorphism having a $2$-sided inverse that is also a ring homomorphism. Isomorphic rings have all their ring-theoretic properties identical. One such ring can be regarded as "the same" as the other.

Definition : Let $R$ and $S$ be rings and let $~φ: R → S~$ be a set map. We say that $~φ~$ is a (ring) isomorphism if
$(1)~ ~φ~$ is a (ring) homomorphism and
$(2)~~ φ~$ is a bijection on sets.

We say that two rings $~R_1~$ and $~R_2~$ are isomorphic if there exists an isomorphism between them. Isomorphic rings differ only by a relabeling of elements.

Notation : $~ R \cong S~.$

Example : If $R$ is a ring, the identity map $~i_d : R → R~$ is an isomorphism of $R$ with itself.

For details find any one of the following:

https://sites.math.washington.edu/~bviray/teaching/RingHomomorphismsAndIsomorphisms.pdf

https://en.wikipedia.org/wiki/Ring_homomorphism#Endomorphisms,_isomorphisms,_and_automorphisms

https://math.okstate.edu/people/binegar/3613/3613-l16.pdf

http://mathonline.wikidot.com/ring-isomorphisms

http://sites.millersville.edu/bikenaga/abstract-algebra-1/ring-maps/ring-maps.pdf

http://campbell.mcs.st-andrews.ac.uk/~john/MT4517/Lectures/L7.html

360 questions
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Purely inseparable extensions vs subextensions

For prime $p$, suppose we have the base field $k = \mathbb{F}_p(x^p,y^p)$ and its extension $\ell = \mathbb{F}_p(x,y)$. How could I prove that for some purely inseparable extension $K/k$ of degree $p$, $K\cong L$ where $\ell/L/k$?
etalic
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Construct a ring isomorphism and its inverse

I have to construct a ring isomorphism $\mathbb{C}[x,y]/\langle x^2+2xy+y^2-529,x+3y-1\rangle \cong \mathbb{C}\times \mathbb C$ and its inverse. I tried to find a function with $\langle x^2+2xy+y^2-529,x+3y-1\rangle$ as a Kernel but couldn't figure…
tom
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When does such isomorphism hold? $\mathbb{Z}/(x^2+5)\cong \mathbb{Z}[\sqrt{-5}]$

When does such isomorphism hold? $$\mathbb{Z}[x]/(x^2+5)\cong \mathbb{Z}[\sqrt{-5}]$$ Edit: Let $R$ be a ring. Consider $p(x)\in R[x]$. Is $R[x]/(p(x)) \cong R[\alpha]$ (where $p(\alpha)=0$) always true? Saw this isomorphism being used many places…
Vinay Deshpande
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Prove that field $\mathbb{Z}[x]/(x,3)$ is isomorphic to field $\mathbb{Z}/3\mathbb{Z}$.

I don't realize what to do. Could you give me some hints or point to solution? I tried to construct surjective homomorphism with $Ker(\phi)=(x, 3)$ by intuition to use First Isomorphism Theorem for Rings, but couldn't find such.
Roma
  • 65
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isomorphism $\mathbb{C} [X,Y]/(X^2 + Y^2 - 1)$

I have to show $\mathbb{C} [X,Y]/(X^2 + Y^2 - 1)$ is isomorphic to $\mathbb{C} [U,V]/(UV - 1)$. First, I would apply the first isomorphism theorem by considering $ \mathbb{C} [X,Y] \rightarrow \mathbb{C} [U,V]/(UV - 1) $. But I'm not sure it will…
Tohiea
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Isomorphic field extenstion

Is $Q(\sqrt5) $ and $Q(\sqrt6)$ isomorphic? Now if I show $$t^2 -5 = (\frac{\sqrt6}{\sqrt5} t)^2 -6=0 $$, does this say they are isomorphic?
user665125
  • 139