# 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.

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### Formula for an inverse isomorphism of commutative rings.

The question I have is styled in the following: "Consider the natural isomorphism of commutative rings:" $\varphi:\mathbb{Z}/(1088) \rightarrow \mathbb{Z}/(17) \oplus \mathbb{Z}/(64)$ Write ...
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### $\mathbb{R}$-Algebra isomorphism $\mathbb{C}[z;\sigma]\cong \mathbb{R} \langle x,y\rangle/(y^2+1,xy+yx)$

Considering the algebra $A=\mathbb{C}[z;\sigma]$ of the skew polynomials, which is like $\mathbb{C}[z]$ but with the multiplication of elements defined the following way: $xb=\sigma(b)x$ and extending ...
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### Testing isomorphism of simple field extensions

Let $F$ be an arbitrary field, $f$ and $g$ irreducible polynomials over $F$. Consider the fields $F[x]/(f(x))$ and $F[y]/(g(y))$. Is there an algorithm to check whether these extensions are isomorphic?...
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### Isomorphism between quotients of two variable formal power series ring

$k$ is a field whose characteristic is not $2$, and $f(x,y)=x^2-y^2, g(x,y)=x^2+x^3-y^2$. Exercise. Show that $k[[x,y]]/(f)\simeq k[[x,y]]/(g)$. So far, I've shown the following. $k[[x,y]]$ is an ...
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### Isomorphism between quotient polynomial rings

Our professor has given us the following isomorphism (fractions denote quotients): $$\frac{\Bbb{Z}[\sqrt{-3}]}{(2+\sqrt{-3})}\cong\frac{\frac{\Bbb{Z}[X]}{(X^2+3)}}{\frac{(2+X,\,X^2+3)}{(X^2+3)}}.$$ ...
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### Isomorphisms in Product Rings

I'm working on a problem in Michael Artin's Algebra that asks: Describe the ring obtained from the product ring $\mathbb{R} \times \mathbb{R}$ by inverting the element $(2,0)$. In the last problem, I ...
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### Is $\mathbb{Q}[1]$ isomorphic to $\mathbb{Q}$?

If I am not mistaken, $\mathbb{Q}[1] \cong \mathbb{Q}/(f(X))$ denotes the ring of polynomials with rational coefficients evaluated in $1$ ($1$ being the root of the polynomial $f(X)$ as well). Could ...
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### Prove that the rings $\frac{\Bbb F_3[X]}{(X^3+X^2+2)}$ and $\frac{\Bbb F_3[X]}{(X^3+2X+2)}$ are isomorphic

Prove that the rings $\frac{\Bbb F_3[X]}{(X^3+X^2+2)}$ and $\frac{\Bbb F_3[X]}{(X^3+2X+2)}$ are isomorphic. I tried to construct an explicit isomorphism using a similar technique as used in this and ...
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There is a theorem in Janusz's Algebraic Number Fields stated as follows: Kummer's Theorem: Let $R$ be a Dedekind ring with quotient field $K$ and $R'$ the integral closure of $R$ in a finite ...